HOME plateWON | World!OfNumbers Palindromic Sums of Powers Wonplate 178

Introduction

Palindromic numbers are numbers which read the same from

left to right (forwards) as from the right to left (backwards)
Here are a few random examples : 7, 3113, 44611644

Palindromic Sums of Powers

The powers expressionThe palindromic curioLength
B.S.Rangaswamy [ June 15, 2009 ] curio 9
605 + 533 + 17027777777779
B.S.Rangaswamy [ July 11, 2009 ] curio 10 - minimum one 3rd, 4th or higher power
230 + 61132 + 172 + 152 + 22111111111110
231 + 86452 + 502 + 72222222222210
577352 + 502 + 242 + 25333333333310
16443 + 10702 + 113 + 152 + 22444444444410
129 + 198942 + 153 + 242 + 24555555555510
18823 + 913 + 1102 + 33666666666610
881912 + 503 + 142 + 102777777777710
233 + 172902 + 104 + 142888888888810
999992 + 4472 + 53 + 26999999999910
B.S.Rangaswamy [ July 11, 2009 ] curio 20
33333333332 + 326 + 61132 + 172 + 152 + 221111111111111111111120
47140452072 + 926422 + 203 + 54 + 222 + 1022222222222222222222220
57735026912 + 1017302 + 463 + 402 + 243333333333333333333320
66666666662 + 942802 + 553 + 632 + 1224444444444444444444420
74535599242 + 1220512 + 4722 + 152 + 1325555555555555555555520
81649658092 + 672872 + 105 + 402 + 636666666666666666666620
88191710362 + 1247252 + 513 + 782 + 1127777777777777777777720
94280904152 + 1243942 + 533 + 202 + 53 + 528888888888888888888820
99999999992 + 1414212 + 105 + 262 + 34
OR
46415883 + 73402262 + 11222 + 182 + 35
9999999999999999999920
```

Dt. 17th July 2009
Dear Mr Patrick

SUB: More Curios

Please find herewith enclosed summation of powers with common base numbers, culminating in palindromes as below:

? 2^41 + 2^34 + - - - - - - - - - -  =  2222222222222 ( 13 Dgts )

? 3^30 + 3^27 + - - - - - - - - - -  =  222222222222222 ( 15 Dgts )

? 5^21 + 5^19 + - - - - - - - - - -  =  555555555555555 ( 15 Dgts )

? 7^14 + 7^12 + - - - - - - - - - -  =  777777777777 ( 12 Dgts )

? 2^25 + 2^10 + - - - - - - - - - -  =  33555533 ( 8 Dgts )

Enclosed is an Excel sheet depicting the above features, for your kind perusal and considerations.

It is felt that palindromes with much larger number of digits can be formulated on similar lines.

With Regards

B.S.Rangaswamy

Note: Sheet1 may be viewed for reference.

1	2		3		5		7		11
2	4		9		25		49		121
3	8		27		125		343		1331
4	16		81		625		2401		14641
5	32		243		3125		16807		161051
6	64		729		15625		117649		1771561
7	128		2187		78125		823543		19487171
8	256		6561		390625		5764801		214358881
9	512		19683		1953125		40353607		2357947691
10	1024		59049		9765625		282475249		25937424601
11	2048		177147		48828125		1977326743		285311670611
12	4096		531441		244140625		13841287201		3138428376721
13	8192		1594323		1220703125		96889010407		34522712143931
14	16384		4782969		6103515625		678223072849		379749833583241
15	32768		14348907		30517578125		4747561509943		4177248169415650	1
16	65536		43046721		152587890625		33232930569601		45949729863572200
17	131072		129140163		762939453125		232630513987207		505447028499294000
18	262144		387420489		3814697265625		1628413597910450		5559917313492230000
19	524288		1162261467		19073486328125		11398895185373100		61159090448414500000
20	1048576		3486784401		95367431640625		79792266297612000		672749994932560000000
21	2097152		10460353203		476837158203125		558545864083284000		7400249944258160000000
22	4194304		31381059609		2384185791015620		3909821048582990000		81402749386839800000000
23	8388608		94143178827		11920928955078100		27368747340080900000		895430243255237000000000
24	16777216		282429536481		59604644775390600		191581231380566000000
25	33554432		847288609443		298023223876953000		1341068619663960000000
26	67108864		2541865828329		1490116119384770000		9387480337647750000000
27	134217728		7625597484987		7450580596923830000		65712362363534300000000
28	268435456		22876792454961		37252902984619100000		459986536544740000000000
29	536870912		68630377364883		186264514923096000000		3219905755813180000000000
30	1073741824		205891132094649		931322574615478000000		22539340290692300000000000
31	2147483648		617673396283947
32	4294967296		1853020188851840
33	8589934592		5559060566555520
34	17179869184		16677181699666600
35	34359738368		50031545098999700
36	68719476736		150094635296999000
37	137438953472		450283905890997000
38	274877906944		1350851717672990000
39	549755813888		4052555153018980000
40	1099511627776		12157665459056900000
41	2199023255552		36472996377170800000
42	4398046511104		109418989131512000000
43	8796093022208		328256967394537000000
44	17592186044416		984770902183611000000
45	35184372088832		2954312706550830000000
46	70368744177664		8862938119652500000000
47	140737488355328		26588814358957500000000
48	281474976710656		79766443076872500000000
49	562949953421312		239299329230618000000000
50	1125899906842620	4	717897987691853000000000
51	2251799813685250	8
52	4503599627370500	496
53	9007199254740990	2
54	18014398509482000	1984
55	36028797018964000	3968
56	72057594037927900	36
57	144115188075856000	5872
58	288230376151712000	1744
59	576460752303423000	488
60	1152921504606850000	46976
61	2305843009213690000	3952
62	4611686018427390000	87904
63
64
65

A	    S q u a r e s

S N	Power	Value
1	2^41	2199023255552
2	2^34	17179869184
3	2^32	4294967296
4	2^30	1073741824
5	2^29	536870912
6	2^26	67108864
7	2^25	33554432
8	2^23	8388608
9	2^22	4194304
10	2^18	262144
11	2^13	8192
12	2^9	512
13	2^8	256
14	2^7	128
15	2^3	8
16	2^2	4
17	2^1	2	*
2222222222222
13 Digit Palindrome

1	2^25	33554432
2	2^10	1024
3	2^6	64
4	2^3	8
5	2^2	4
6	2^0	1	*
33555533
8 Digit Palindrome
OR

1	2^25	33554432
2	10^3	1000
3	10^2	100
4	10^0	1	*
33555533
8 Digit Palindrome

B	C u b e s

S N	Power	Value
1	3^30	205891132094649
2	3^27	7625597484987
3	3^27	7625597484987
4	3^25	847288609443
5	3^23	94143178827
6	3^23	94143178827
7	3^22	31381059609
8	3^21	10460353203
9	3^19	1162261467
10	3^19	1162261467
11	3^17	129140163
12	3^15	14348907
13	3^14	4782969
14	3^14	4782969
15	3^12	531441
16	3^12	531441
17	3^10	59049
18	3^10	59049
19	3^8	6561
20	3^8	6561
21	3^7	2187
22	3^7	2187
23	3^6	729
24	3^5	243
25	3^5	243
26	3^3	27
27	3^3	27
28	3^1	3
222222222222222
15 Digit palindrome

*	Power < 2

C	5th Powers

S N	Power	Value
1	5^21	476837158203125
2	5^19	19073486328125
3	5^19	19073486328125
4	5^19	19073486328125
5	5^19	19073486328125
6	5^17	762939453125
7	5^17	762939453125
8	5^17	762939453125
9	5^15	30517578125
10	5^15	30517578125
11	5^15	30517578125
12	5^15	30517578125
13	5^14	6103515625
14	5^14	6103515625
15	5^13	1220703125
16	5^11	48828125
17	5^11	48828125
18	5^10	9765625
19	5^10	9765625
20	5^10	9765625
21	5^9	1953125
22	5^9	1953125
23	5^9	1953125
24	5^9	1953125
25	5^8	390625
26	5^8	390625
27	5^7	78125
28	5^5	3125
29	5^5	3125
30	5^4	625
31	5^4	625
32	5^4	625
33	5^3	125
34	5^3	125
35	5^3	125
36	5^3	125
37	5^2	25
38	5^2	25
39	5^1	5
555555555555555
15 Digit palindrome

D	7th Powers

S N	Power	Value
1	7^14	678223072849
2	7^12	96889010407
3	7^11	1977326743
4	7^10	282475249
5	7^10	282475249
6	7^9	40353607
7	7^9	40353607
8	7^9	40353607
9	7^7	823543
10	7^7	823543
11	7^6	117649
12	7^6	117649
13	7^6	117649
14	7^6	117649
15	7^6	117649
16	7^6	117649
17	7^4	2401
18	7^3	343
19	7^3	343
20	7^3	343
21	7^2	49
777777777777
12 Digit Palindrome

Dt. 25th July 2009
Dear Mr Patrick

SUB: Few More Curios

In continuation of my submissions dated 17th July, please find herewith

enclosed details of Repdigital Palindromes together with their constituents for

11th,13th, 17th & 19th  powers for your kind perusal and considerations.

With Regards

B.S.Rangaswamy

Dt.25.07.2009
Dear Mr Patrick

SUB: Few more Curios

E   4444444444444444 (16 Digits) - 13

= 11^15 + 7(11^13) + 8(11^12) + 11^11 + 5(11^10) + 6(11^9) + 3(11^8)

+ 3(11^7) + 8(11^6) + 5(11^5) + 8(11^3) + 9(11^2) + 10(11^1)

F   222222222222 (12 Digits) - 9

=  13^10 + 7(13^9) + 12(13^8) + 5 (13^7) + 6(13^6) + 2(13^5)

+ 9(13^3) + 13^2 + 11(13^1)

G   9999999999999999 (16 Digits) - 11

=   17^13 + 2(17^11) + 13(17^10) + 5(17^9) + 11(17^8) + 17^7

+ 7(17^6) + !7^5 + 4(17^4) + 10(17^2) + 13(17^1)

H   888888888888888888 (18 Digits) - 14

=   19^14 + 2(19^13) + 2(19^12) + 11(19^11) + 11(19^10) + 2(19^9)

+ 7(19^8) + 8(19^7) + 7(19^6) + 10(19^5) + 9(19^4) + 11(19^3)

+ 2(19^2) + 7(19^1)

Following are revised details of  Repdigital palindromes, submitted earlier:

A   2222222222222 (13 Digits) - 17

=    2^41 + 2^34 + 2^32 + 2^30 + 2^29 + 2^26 + 2^25 + 2^23 + 2^22 + 2^18

+ 2^13 + 2^9 + 2^8 + 2^7 + 2^3 + 2^2 + 2^1

B   222222222222222 (15 Digits) - 18

=   3^30 + 2(3^27) + 3^25 + 2(3^23) + 3^22 + 3^21 + 2(3^19) + 3^17 + 3^15

+2(3^14) + 2(3^12) + 2(3^10) + 2(3^8) + 2(3^7) + 3^6 + 2(3^5) + 2(3^3)

+ 3^1

C  555555555555555 (15 Digits) - 16

=   5^21 + 4(5^19) + 3(5^17) + 4(5^15) + 3(5^14) + 5^13 + 2(5^11) + 2(5^10)

+ 4(5^9) + 2(5^8) + 5^7 + 2(5^5) + 2(5^4) + 4(5^3) + 2(5^2) + 5^1

D  777777777777 (12 Digits) - 10

=  7^14 + 7^12 + 7^11 + 2(7^10) + 3(7^9) + 2(7^7) + 6(7^6) + 7^4 + 3(7^3)

+ 7^2

Please see Enclosed Excel sheets 2 & 1 for details. Excel tables are now

revised for Cubes, 5th & 7th powers.

With Regards

B.S.Rangaswamy

11	13	17	19

121	169	289	361
1331	2197	4913	6859	11
14641	28561	83521	130321	9
161051	371293	1419857	2476099	10
1771561	4826809	24137569	47045881	7
19487171	62748517	410338673	893871739	8
214358881	815730721	6975757441	16983563041	7
2357947691	10604499373	118587876497	322687697779	2
25937424601	137858491849	2015993900449	6131066257801	11
285311670611	1792160394037	34271896307633	116490258898219	11
3138428376721	23298085122481	582622237229761	2213314919066160	1	2
34522712143931	302875106592253	9904578032905940	42052983462257100	0	59	2
379749833583241			799006685782884000		121
4177248169415650
45949729863572200
505447028499294000
5559917313492230000
61159090448414500000
672749994932560000000
7400249944258160000000
81402749386839800000000
895430243255237000000000

A	    S q u a r e s

S N	Power	Value
1	2^41	2199023255552
2	2^34	17179869184
3	2^32	4294967296
4	2^30	1073741824
5	2^29	536870912
6	2^26	67108864
7	2^25	33554432
8	2^23	8388608
9	2^22	4194304
10	2^18	262144
11	2^13	8192
12	2^9	512
13	2^8	256
14	2^7	128
15	2^3	8
16	2^2	4
17	2^1	2	*
2222222222222
13 Digit Palindrome

1	2^25	33554432
2	2^10	1024
3	2^6	64
4	2^3	8
5	2^2	4
6	2^0	1	*
33555533
8 Digit Palindrome
OR
1	2^25	33554432
2	10^3	1000
3	10^2	100
4	10^0	1
33555533	*
8 Digit Palindrome

B	C u b e s

S N	Power	Value
1	3^30	205891132094649
2	2(3^27)	15251194969974
3	3^25	847288609443
4	2(3^23)	188286357654
5	3^22	31381059609
6	3^21	10460353203
7	2(3^19)	2324522934
8	3^17	129140163
9	3^15	14348907
10	2(3^14)	9565938
11	2(3^12)	1062882
12	2(3^10)	118098
13	2(3^8)	13122
14	2(3^7)	4374
15	3^6	729
16	2(3^5)	486
17	2(3^3)	54
18	3^1	3	*
222222222222222
15 Digit palindrome

*	Power < 2

C	5th Powers

S N	Power	Value
1	5^21	476837158203125
2	4(5^19)	76293945312500
3	3(5^17)	2288818359375
4	4(5^15)	122070312500
5	2(5^14)	12207031250
6	5^13	1220703125
7	2(5^11)	97656250
8	2(5^10)	29296875
9	4(5^9)	7812500
10	2(5^8)	781250
11	5^7	78125
12	2(5^5)	6250
13	3(5^4)	1875
14	4(5^3)	500
15	2(5^2)	50
16	5^1	5	*
555555555555555
15 Digit palindrome

D	7th Powers

S N	Power	Value
1	7^14	678223072849
2	7^12	96889010407
3	7^11	1977326743
4	2(7^10)	564950498
5	3(7^9)	121060821
6	2(7^7)	1647086
7	6(7^6)	705894
8	7^4	2401
9	3(7^3)	1029
10	7^2	49
777777777777
12 Digit Palindrome

Dt. 29.07.2009
Dear Mr Patrick

Thank you immensely for calling me to participate in this

highly interesting and educating Topic. I have used more than one square or

cube or other powers in each of my findings, which may not match your ultimate

requirement. It is for others to explore and arrive at such high value palindromes.

With Regards

B.S.Rangaswamy

C U R I O  -  32
(32 Digit palindromes)

11111111111111111111111111111111  (7,7)

=  3333333333333333^2 + 47140452^2 +2730^2 + 17^3 + 3^4 + 2^4 + 2^3

OR

=  165^14 + 158437088969584^2 + 16236309^2 + 5051^2 + 18^3 + 2^9 + 2^2

22222222222222222222222222222222  (6,7)

=  4714045207910316^2 + 88425577^2 + 7430^2 + 120^2 + 11^2 + 2^4*

OR

=  8286^8 + 37887508351616^2 + 8195697^2 + 221^3 + 38^3 + 52^ 2+ 2^6

33333333333333333333333333333333 (7,7)

=  5773502691896257^2 + 86306864^2 + 10450^2 + 79^2 + 3^3 + 2^4 + 2^2

OR

=  734^11 + 130542669028054^2 + 6563007^2 + 217^3 + 45^3 + 85^2+ 11^2

44444444444444444444444444444444 (7,7)

=  6666666666666666^2 + 94280904^2 + 5461^2 + 19^3 + 46^2 +12^2  + 3^5

OR

=  9036^8 + 28078249904977^2 + 5208833^2 + 101^3 + 13^3 + 2^6 + 2^3

55555555555555555555555555555555 (6,6)

=  7453559924999298^2 + 121361815^2 + 10722^2 +85^2 + 3^2 + 2^3

OR

=  7453559924999298^2 + 121361815^2 + 486^3 + 421^2 +5^2 + 2^2

66666666666666666666666666666666 (6,7)

=  8164965809277260^2 + 73110759^2 + 5657^2 + 6^4 + 2^5 + 2^3

OR

=  8164965809277260^2 + 73110759^2 + 317^3 + 52^3 + 19^3

+17^2 + 6^3

77777777777777777777777777777777 (6,7)

=  8819171036881968^2 + 105832141^2 + 526^3 + 222^2 + 2^3 + 2^2

OR

=  33^21 + 607082836486390^2 + 19150661^2 + 261^3 + 17^2 + 7^2+ 2^2

88888888888888888888888888888888 (6,7)

=  9428090415820633^2 + 111445726^2 + 14720^2 + 150^2 + 14^2 + 3^3

OR

=  9428090415820633^2 + 111445726^2 + 14720^2 + 28^3 + 2^9 + 3^5+ 2^4

99999999999999999999999999999999 (7,6,6)

=  9999999999999999^2 + 141421356^2 + 406^3 + 58^3 + 13^3 + 2^9 + 5^2

OR

=  9999999999999999^2 + 109^8 + 8624012^2 + 349^2 + 10^2 + 2^5

OR

=  46415888336^3 + 28739148791^2 +141768^2 + 442^2 + 7^2 + 5^2

Following is a random palindrome, dictated by my grandson over telephone:

37985621462109866890126412658973 (5,6)

Its constituents are

6163247639200445^2 + 90681241^2 + 7427^2 + 23^2 + 3^2

OR

33615513156^3 + 47496382543^2 + 271934^2 + 715^2 + 10^2 + 3^3

OR

-    -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -   -

With the liberty to use Squares, Cubes and other powers more than once,

Every palindrome can be expressed as summation of powers in several ways!!

To arrive at the least number of constituent powers is really is an intellectual task.

- B.S.Rangaswamy

> From: patrick.degeest@skynet.be
> To: psdevices@hotmail.com
> Subject: RE: C U R I O 32
> Date: Mon, 3 Aug 2009 01:27:56 +0200
>
> Dear B.S.Rangaswamy,
>
> Thanks a lot for the latest submissions regarding the repdigits
> expressed in a variety of sums of powers. They are indeed very
> nice but alas are not directly solutions as I intended them
> for wonplate 178. The reason is that I think they are
> 'engineered' rather than 'sporadic'. In that spirit the
> sumpower of your grandson is more like it and I will add
> his palindrome to the plate.
>
> Of course I wouldn't like to lose all those nice repdigital
> palindromes that you send in. Instead I will create a dedicated
> but separate page for these numbers, in due time. Please be
> patient and all will come to a good end. As soon as it is finished
>
> Best regards,
>
> P@rick.
>
> Patrick De Geest
> 1. mailto:pdg@www.worldofnumbers.com
> 2. mailto:Patrick.DeGeest@skynet.be
> website 1 : http://www.worldofnumbers.com/index.html
> website 2 : http://users.skynet.be/worldofnumbers/ (mirrorsite)
>
>
> > Van: B.S.Rangaswamy . [mailto:psdevices@hotmail.com]
> > Verzonden: woensdag 29 juli 2009 10:07
> > Aan: patrick DeGeest
> > CC: psdevices@hotmail.com
> > Onderwerp: C U R I O 32
> >
> > Dear Mr Patrick
> >
> >        Kindly see enclosures
> >
> >
> > B.S.Rangaswamy
> >
> > Dear Mr Patrick

In thanking for your earnest feed back on my findings and inclusion of 32 digit palindrome of my grandson together with its constituents,you may take appropriate action on my other findings in this regard .

With Regards

B.S.Rangaswamy

Dear Mr Patrick

Further to my earlier reply, I definately agree that all the findings submitted by me are engineered, and not spariadic as desired/pronounced. Regret for diversifying on this very
interesting subject topic.

B.S.R

Dt. 6th August 2009
Dear Mr Patrick

Kindly excuse me for disturbing and drawing your attention to a few of my remaining findings on some more repdigit palindromes generated from powers of base numbers 23, 29, 31, 37, 41, 43, 53, 73 and 79 vide enclosures for your kind perusal. It is true that these are not sporadic, but engineered for varied variety of exploration.

I only wish we can come across some very enchanting palindromes with many scores of digits as sum of powers, through your renowned website
World! of Numbers.

With Regards

B.S.Rangaswamy

Repdigit Palindromes as Sum of powers of primes

B.S.Rangaswamy

1111111111111111111111 (22 Digits) - 14 Constituents

=  4(23^15) + 3(23^14) + 19(23^13) + 9(23^12) + 18(23^11)  +14(23^10)

+ 4(23^9)  + 9(23^7) + 8(23^6) + 23^5 + 16(23^4) + 8(23^3) + 2(23^2)

+ 6(23^1).

1111111111111111111111111111 (28 Digits) - 18 Constt.

= 5(29^18) + 8(29^17) + 3(29^16) + 29^15 + 27(29^14) + 10(29^13) + 5(29^12)

+ 27(29^11) + 10(29^10) + 24(29^9) + 17(29^8) + 27(29^7) + 24(29^6)

+ 18(29^5) + 28(29^4) + 11(29^3) + 2(29^2) + 24(29^1).

333333333333333 (15 Digits) - 9 Constt.

= 12(31^9)  + 18(31^8) + 25(31^7)  + 20(31^6)  + 9(31^5)  + 6(31^4)

+ 3(31^3)  + 12(31^2)  +18(31^1).

666666666666666 (15 Digits)  9 Constt.

=  5(37^9) + 4(37^8) + 29(37^7) + 21(37^6) + 16(37^5) + 19(37^4)

+ 37^3 + 15(37^2) + 16(37^1).

333333333333333 (15 Digits)  8 Constt.

=  41^9 + 30(41^7) + 22(41^6) + 36(41^5) + 34(41^4) + 22(41^3)

+ 8(41^2) + 20(41^1).

333333333333333333333 (21 Digits)  12 Constt.

=  8(43^12) + 14(43^11) + 29(43^10) + 38(43^9) + 29(43^8) + 15(43^7)

+ 9(43^6) + 29(43^5) + 15(43^4) + 34(43^3) + 25(43^2) + 4(43^1).

9999999999999 (13 Digits)  7 Constt.

=  8(53^7) + 27(53^6) + 9(53^5) + 13(53^4)  + 45(53^3) + 34(53^2)

+ 3(53^1).

3333333333333333 (16 Digits)  8 Constt.

=  4(73^8) + 9(73^7) + 53(73^6) + 22(73^5) + 2(73^4)

+ 73^3 + 62(73^2) + 12(73^1)

2222222222222 (13 Digits)  6 Constt.

=  9(79^6) + 11(79^5)  + 15(79^4) + 6(79^3)  + 46(79^2) + 71(79^1).

- - - - - - - - - - - - - - -

Note:  No repdigit palindromes of 32 digits and below, can be generated from
powers of primes  47, 59. 61, 67, 71, 83, 89 & 97 unless K(N^0) is used at
the end, in each of these cases similar to generation of 8 digit palindrome
33555533 under powers of Two.

Excel sheets for generating repdigital palindromes for powers of
2, 3, 5, 7, 11, 13, 17, 19, 31, 37, 41, 53, 73 and 79 are furnished at
sheet 2 of enclosure - P O W R.

- - - - - - - - - - - - - - - - - - - -

59	73	79
3481	5329	6241
205379	389017	493039
12117361	28398241	38950081
714924299	2073071593	3077056399
42180533641	151334226289	243087455521
2488651484819	11047398519097	19203908986159
146830437604321	806460091894081	1517108809906560	1
33	8662995818654940	58871586708267900	13

Sum of Powers																			        Sum of Powers														        Sum of Powers												        Sum of Powers
A	Powers of TWO			B	Powers of THREE			C	Powers of FIVE			D	Powers of SEVEN						E	Powers of 11			F	Powers of 13			G	Powers 0f 17				H	Powers of 19				I	Powers of  31			J	Powers of 37			K	Powers of 41			L	Powers of 53			M	Powers of 73				N	Powers of 79
S N	Power	Value		S N	Power	Value		S N	Power	Value		S N	Power	Value					S N	Power	Value	+	S N	Power	Value		S N	Power	Value	+		S N	Power	Value	+		S N	Power	Value		S N	Power	Value		S N	Power	Value		S N	Power	Value		S N	Power	Value	+		S N	Power	Value
1	2^41	2199023255552		1	3^30	205891132094649		1	5^21	476837158203125		1	7^14	678223072849					1	11^15	4177248169415650	1	1	13^10	137858491849		1	17^13	9904578032905940	7		1	19^14	799006685782884000	121		1	12(31^9)	317275465928052		1	5(37^9)	649808698975385		1	41^9	327381934393961		1	8(53^7)	9397689118696		1	4(73^8)	3225840367576320	4		1	9(79^6)	2187787099689
2	2^34	17179869184		2	2(3^27)	15251194969974		2	4(5^19)	76293945312500		2	7^12	96889010407					2	7(11^13)	241658985007510	7	2	7(13^9)	74231495611		2	2(17^11)	68543792615260	6		2	2(19^13)	84105966924514000	118		2	18(31^8)	15352038673938		2	4(37^8)	14049917815684		2	30(41^7)	5842628216430		2	27(53^6)	598437750483		2	9(73^7)	99426586671870	3		2	11(79^5)	33847620389
3	2^32	4294967296		3	3^25	847288609443		3	3(5^17)	2288818359375		3	7^11	1977326743					3	8(11^12)	25107427013760	8	3	12(13^8)	9788768652		3	13(17^10)	26207920705830	7		3	2(19^12)	4426629838132000	322		3	25(31^7)	687815352775		3	29(37^7)	2753024436857		3	22(41^6)	104502293302		3	9(53^5)	3763759437		3	53(73^6)	8020713993310	7		3	15(79^4)	584251215
4	2^30	1073741824		4	2(3^23)	188286357654		4	4(5^15)	122070312500		4	2(7^10)	564950498					4	11^11	285311670610	1	4	5(13^7)	313742585		4	5(17^9)	592939382480	5		4	11(19^11)	1281392847880000	409		4	20(31^6)	17750073620		4	21(37^6)	53880254589		4	36(41^5)	4170823236		4	13(53^4)	102576253		4	22(73^5)	45607575040	6		4	6(79^3)	2958234
5	2^29	536870912		5	3^22	31381059609		5	2(5^14)	12207031250		5	3(7^9)	121060821					5	5(11^10)	129687123000	5	5	6(13^6)	28960854		5	11(17^8)	76733331850	1		5	11(19^10)	67441728835000	811		5	9(31^5)	257662359		5	16(37^5)	1109503312		5	34(41^4)	96075874		5	45(53^3)	6699465		5	2(73^4)	56796480	2		5	46(79^2)	287086
6	2^26	67108864		6	3^21	10460353203		6	5^13	1220703125		6	2(7^7)	1647086					6	6(11^9)	14147686140	6	6	2(13^5)	742586		6	17^7	410338670	3		6	2(19^9)	645375395000	558		6	6(31^4)	5541126		6	19(37^4)	35609059		6	22(41^3)	1516262		6	34(53^2)	95506		6	73^3	389010	7		6	71(79^1)	5609	*
7	2^25	33554432		7	2(3^19)	2324522934		7	2(5^11)	97656250		7	6(7^6)	705894					7	3(11^8)	643076640	3	7	9(13^3)	19773		7	7(17^6).	168962890	3		7	7(19^8)	118884941000	287		7	3(31^3)	89373		7	37^3	50653		7	8(41^2)	13448		7	3(53^1)	159	*	7	62(73^2)	330390	8				2222222222222
8	2^23	8388608		8	3^17	129140163		8	2(5^10)	29296875		8	7^4	2401					8	3(11^7)	58461510	3	8	13^2	169		8	17^5	1419850	7		8	8(19^7)	7150973000	912		8	12(31^2)	11532		8	15(37^2)	20535		8	20(41^1)	820	*			9999999999999		8	12(73^1)	870	6	*			13 Digit palindrome
9	2^22	4194304		9	3^15	14348907		9	4(5^9)	7812500		9	3(7^3)	1029					9	8(11^6)	14172480	8	9	11(13^1)	143	*	9	4(17^4)	334080	4		9	7(19^6)	329321000	167		9	18(31^1)	558	*	9	16(37^1)	592	*			333333333333333				13 Digit Palindrome				3333333333333290	43
10	2^18	262144		10	2(3^14)	9565938		10	2(5^8)	781250		10	7^2	49					10	5(11^5)	805250	5			222222222222		10	10(17^2)	2890	0		10	10(19^5)	24760000	990				333333333333333				666666666666666				15 Digit Palindrome								16 Digit palindrome
11	2^13	8192		11	2(3^12)	1062882		11	5^7	78125				777777777777					11	8(11^3)	10640	8			12 Digit Palindrome		11	13(17^1)	210	6	*	11	9(19^4)	1172000	889				15 Digit Palindrome				15 Digit Palindrome
12	2^9	512		12	2(3^10)	118098		12	2(5^5)	6250				12 Digit Palindrome					12	9(11^2)	1080	9							9999999999999950	49		12	11(19^3)	75000	449
13	2^8	256		13	2(3^8)	13122		13	3(5^4)	1875									13	10(11^1)	110	0	*						     16 Digit Palindrome			13	2(19^2)	0	722
14	2^7	128		14	2(3^7)	4374		14	4(5^3)	500											4444444444444380	64										14	7(19^1)	0	133	*
15	2^3	8		15	3^6	729		15	2(5^2)	50											    16 Digit Palindrome													888888888888882000	6888
16	2^2	4		16	2(3^5)	486		16	5^1	5	*																							         18 Digit Palindrome
17	2^1	2	*	17	2(3^3)	54				555555555555555
2222222222222		18	3^1	3	*			15 Digit palindrome
13 Digit Palindrome				222222222222222
15 Digit palindrome

1	2^25	33554432			*	Power < 2
2	2^10	1024
3	2^6	64
4	2^3	8
5	2^2	4
6	2^0	1	*
33555533
8 Digit Palindrome
OR
1	2^25	33554432
2	10^3	1000
3	10^2	100
4	10^0	1	*
33555533
8 Digit Palindrome

Repdigital palindromes as sum of multi powers of 101

The second lowest Prime Palindrome 101 is an exceptional number, in the Numerical World. Its multiple powers can add up to innumerable Number of Repdigit palindromes (RDP) of digits, which are multiples of 4.

All RDPs of  8.12,16 digits together with their multiple powers of 101
are enclosed at Excel sheet 2 of POW 101.

The fifteen constituents of  RDP

11111111111111111111111111111111 (32 Digits)

are:        9(101^15)
57(101^14)
63(101^13)
17(101^12)
95(101^11)
69(101^10)
33(101^9)
48(101^8)
39(101^7)
43(101^6)
32 (101^5)
89(101^4)
39(101^3)
81(101^2)
88(101^1).

RDPs of 32 digits starting from 2, 3, 4, 5, 6, 7, 8 and 9 can all be
expressed as sum of multiple powers of 101 in similar ways. RDPs of
36, 40, 44 - - - - digits can also be expressed as sum of multiple powers
of 101. This phenomenon has no limits and can go on & on to infinite
limits of the number of digits of RDPs in the World of Numbers!!

- B.S.Rangaswamy

Dt. 8th August 2009

4
3	107213535210701	62	6647239183063460	2
2	1061520150601	18	19107362710810	8
1	10510100501	30	315303015030	0
104060401	46	4786778440	6
1030301	30	30909030	0
5	10201	18	183610	8
4	101	62	6260	2
3			6666666666666640	26
2
1	107213535210701	72	7719374535170470	2
1061520150601	55	58383608283050	5
10510100501	1	10510100500	1
5	104060401	87	9053254880	7
4	1030301	68	70060460	8
3	10201	89	907880	9
2	101	5	500	5
1			7777777777777740	37

107213535210701	82	8791509887277480	2
5	1061520150601	91	96598333704690	1
4	10510100501	74	777747437070	4
3	104060401	28	2913691220	8
2	1030301	6	6181800	6
1	10201	58	591650	8
101	49	4940	9
8888888888888850	38
5
4	107213535210701	93	9970858774595190	3
3	1061520150601	27	28661044066220	7
2	10510100501	45	472954522540	5
1	104060401	69	7180167660	9
1030301	45	46363540	5
10201	27	275420	7
5	101	93	9390	3
4			9999999999999960	39

8 Digit Palindromes					      12 Digit Palindromes				          16 Digit Palindromes						          16 Digit Palindromes

S N	Power	Value			S N	Power	Value		S N	Power	Value	+			S N	Power	Value	+
1	10(101^3)	10303010			1	10(101^5)	105101005010		1	10(101^7)	1072135352107010	0			1	62(101^7)	6647239183063460	2
2	79(101^2)	805879			2	57(101^4)	5931442857		2	36(101^6)	38214725421630	6			2	18(101^6)	19107362710810	8
3	22(101^1)	2222			3	76(101^3)	78302876		3	72(101^5)	756727236070	2			3	30(101^5)	315303015030	0
Total	11111111			4	35(101^2)	357035		4	41(101^4)	4266476440	1			4	46(101^4)	4786778440	6
5	33(101^1)	3333		5	38(101^3)	39151430	8			5	30(101^3)	30909030	0
1	21(101^3)	21636321				Total	111111111111		6	70(101^2)	714070	0			6	18(101^2)	183610	8
2	57(101^2)	581457							7	44(101^1)	4440	4			7	62(101^1)	6260	2
3	44(101^1)	4444			1	21(101^5)	220712110521			Total	1111111111111090	21				Total	6666666666666640	26
Total	22222222			2	14(101^4)	1456845614
3	51(101^3)	52545351		1	20(101^7)	2144270704214020	0			1	72(101^7)	7719374535170470	2
1	32(101^3)	32969632			4	70(101^2)	714070		2	73(101^6)	77490970993870	3			2	55(101^6)	58383608283050	5
2	35(101^2)	357035			5	66(101^1)	6666		3	43(101^5)	451934321540	3			3	101^5	10510100500	1
3	66(101^1)	6666				Total	222222222222		4	82(101^4)	8532952880	2			4	87(101^4)	9053254880	7
Total	33333333							5	77(101^3)	79333170	7			5	68(101^3)	70060460	8
1	31(101^5)	325813115531		6	39(101^2)	397830	9			6	89(101^2)	907880	9
1	43(101^3)	44302943			2	72(101^4)	7492348872		7	101*	8880	8			7	5(101^1)	500	5
2	13(101^2)	132613			3	27(101^3)	27818127			*101	2222222222222190	32				Total	7777777777777740	37
3	88(101^1)	8888			4	4(101^2)	40804			*
Total	44444444			5	99(101^1)	9999		1	*101	3323619591531730	1			1	82(101^7)	8791509887277480	2
Total	333333333333		2	9(101^6)	9553681355400	9			2	91(101^6)	96598333704690	1
1	53(101^3)	54605953							3	15(101^5)	157851507510	5			3	74(101^5)	777747437070	4
2	93(101^2)	948693			1	42(101^5)	441424221042		4	23(101^4)	2393389220	3			4	28(101^4)	2913691220	8
3	9(101^1)	909			2	29(101^4)	3017751629		5	15(101^3)	15454510	5			5	6(101^3)	6181800	6
Total	55555555			3	2(101^3)	2060602		6	9(101^2)	91800	9			6	58(101^2)	591650	8
4	40(101^2)	408040		7	31(101^1)	3130	1			7	49(101^1)	4940	9
1	64(101^3)	65939264			5	31(101^1)	3131			Total	3333333533333300	33				Total	8888888888888850	38
2	71(101^2)	724271				Total	444444444444
3	31(101^1)	3131							1	41(101^7)	4395754943638740	1			1	93(101^7)	9970858774595190	3
Total	66666666			1	52(101^5)	546525226052		2	45(101^6)	47768406777040	5			2	27(101^6)	28661044066220	7
2	86(101^4)	8949194486		3	87(101^5)	914378743580	7			3	45(101^5)	472954522540	5
1	75(101^3)	77272575			3	78(101^3)	80363478		4	64(101^4)	6659865660	4			4	69(101^4)	7180167660	9
2	49(101^2)	499849			4	75(101^2)	765075		5	53(101^3)	54605950	3			5	45(101^3)	46363540	5
3	53(101^1)	5353			5	64(101^1)	6464		6	79(101^2)	805870	9			6	27(101^2)	275420	7
Total	77777777				Total	555555555555		7	75(101^1)	7570	5			7	93(101^1)	9390	3
Total	4444444444444410	34				Total	9999999999999960	39
1	86(101^3)	88605886			1	63(101^5)	662136331563
2	27(101^2)	275427			2	43(101^4)	4474597243		1	51(101^7)	5467890295745750	1
3	75(101^1)	7575			3	54(101^3)	55636254		2	82(101^6)	87044652349280	2
Total	88888888			4	9(101^2)	91809		3	59(101^5)	620095929550	9
5	97(101^1)	9797		4	4(101^4)	416241600	4
1	97(101^3	99939197				Total	666666666666		5	92(101^3)	94787690	2
2	5(101^2)	51005							6	49(101^2)	499840	9
3	97(101^1)	9797			1	74(101^5)	777747437074		7	18(101^1)	1810	8
Total	99999999			2	29(101^3)	29878729			Total	5555555555555520	35
3	45(101^2)	459045
4	29(101^1)	2929
Total	777777777777

1	84(101^5)	882848442084
2	58(101^4)	6035503258
3	4(101^3)	4121204
4	80(101^2)	816080
5	62(101^1)	6262
Total	888888888888

1	95(101^5)	998459547595
2	14(101^4)	1456845614
3	81(101^3)	83454381
4	14(101^2)	142814
5	95(101^1)	9595
Total	999999999999

10th August 2009
Dear Mr Patrick

Please find enclosed revised page 1 0 1 - MPR. This is in place of 1 0 1 - MP, submitted earlier, on 6th August. The only revisions are:

A. Serial numbers are added to the list of multipowers of 101.

B. "infinite range" replaces the earlier "infinite limits" in the last paragraph.

Thanking You

B.S.Rangaswamy

Dt. 15th August 2009
Dear Mr Patrick

Please find enclosed corrected multi powers of 101 for generating RDP of
32 digits starting with 1, vide 1 0 1 MPR for your kind perusal. Extremely sorry for the error committed earlier.

This is now prepared in Excel sheet. All other 32 digit RDPs starting with
2, 3, 4 - - - -9  are also being prepared on Excel. I shall be submitting these to you on completion.

Thanking you ,

With Regards

B.S.Rangaswamy

Repdigital palindromes as sum of multi powers of 101

The second lowest Prime Palindrome 101 is an exceptional number, in the Numerical World. Its multiple powers can add up to innumerable Number of Repdigit palindromes (RDP) of digits, which are multiples of 4.

All RDPs of  8.12,16 digits together with their multiple powers of 101
are enclosed at Excel sheet 2 of POW 101.

The fifteen constituents of  RDP

11111111111111111111111111111111 (32 Digits)

are:        1.       9(101^15)
2.      57(101^14)
3.      63(101^13)
4.      17(101^12)
5.      95(101^11)
6.      69(101^10)
7.      33(101^9)
8.      48(101^8)
9.      39(101^7)
10.      43(101^6)
11.      32(101^5)
12.      89(101^4)
13.      38(101^3)*
14.      91(101^2)*        * Corrected powers
15.      88(101^1)

RDPs of 32 digits starting from 2, 3, 4, 5, 6, 7, 8 and 9 can all be
expressed as sum of multiple powers of 101 in similar ways. RDPs of
36, 40, 44 - - - - digits can also be expressed as sum of multiple powers
of 101. This phenomenon has no limits and can go on & on to infinite
range of the number of digits of RDPs in the World of Numbers!!

- B.S.Rangaswamy

Dt.15th August 2009

Dt. 26th August 2009
Dear Mr Patrick

Kindly excuse me for this belated submission. In continuation of my previous mails, please find herewith enclosed RPDT P, an Excel format of Sum of multiple powers of 101 generating Repdigital palindromes of 32 digits starting with 1, 2, .. . .to.9,  for your kind perusal. These are not sporadic, but engineered to derive repdigital palindromes. For your kind information sheet 1 of this format is protected where as sheet 2 is unprotected. You can make use of sheet 2 for any likely additions / deletions.

Your encouraging statements terming Nine sevens as Gem; appreciating
my patience at certain stage, have resulted in arriving patiently at these results.

Every Repdigital palindrome, whose number of digits is a multiple
of 4 can be expressed as sum of multiple powers of 101 the second lowest  palindrome prime in the World of Numbers. This phenomenon can extend even
up to infinite range!

The random palindrome of 32 digits is expressed in three different ways, which is illustrated in sheet 3 of Excel. At C this palindrome is generated from multiple powers of the lowest palindrome prime 11.

I am highly grateful to you for bestowing me this opportunity to participate in this very interesting and educative topic, through your esteemed website.

Thanking You,

With Regards

B.S.Rangaswamy

S U M   O F   M U L T I P L E   P O W E R S   O F   " 1 0 1 "

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	9(101^15)	=	104487205983	299868165026	O4463509					1	38(101^15)	=	441168203040	599443363443	29957038		1	66(101^15)	=	766239510544	199033210190	99399066
2	57(101^14)	=	6552003015	454447178664	99949857					2	28(101^14)	=	3218527797	O65342473730	17519228		2	100(101^14)	=	11494742132	376223120464	91140100
3	63(101^13)	=	71699876	667297233325	67221963					3	50(101^13)	=	56904664	O21664470893	39065050		3	38(101^13)	=	43247544	656464997878	97689438
4	17(101^12)	=	191560	255122434852	51240417					4	71(101^12)	=	800045	771393698501	66945271		4	24(101^12)	=	270438	O07231672732	95868824
5	95(101^11)	=	10598	849293320507	27354595					5	79(101^11)	=	8813	779938656000	78536979		5	63(101^11)	=	7028	710583991494	29719363
6	69(101^10)	=	76	218926653373	11119069					6	75(101^10)	=	82	846659405840	33825075		6	81(101^10)	=	89	474392158307	56531081
7	33(101^9)	=		360916139985	83909733					7	32(101^9)	=		349979287258	99548832		7	32(101^9)	=		349979287258	99548832
8	48(101^8)	=		5197712187	O1478448					8	92(101^8)	=		9962281691	77833692		8	35(101^8)	=		3789998469	69828035
9	39(101^7)	=		41813278	73217339					9	56(101^7)	=		60039579	71799256		9	74(101^7)	=		79338016	O5591874
10	43(101^6)	=		456453	66475843					10	72(101^6)	=		764294	50843272		10	28(101^5)	=		2942	82814028
11	32(101^5)	=		3363	23216032					11	30(101^5)	=		3153	O3015030		11	19(101^4)	=		19	77147619
12	89(101^4)	=		92	61375689					12	54(101^4)	=		56	19261654		12	70(101^3)	=			72121070
13	38(101^3)	=			39151438					13	54(101^3)	=			55636254		13	37(101^2)	=			377437
14	91(101^2)	=			928291					14	64(101^2)	=			652864		14	10(101^1)	=			1010
15	88(101^1)	=			8888					15	49(101^1)	=			4949					777777777775	2770546105037	872185903
111111111108	3111111111105	605169154								444444444441	3357437499815	641429414					777777777777	777777777777	77777777
111111111111	111111111111	11111111								444444444444	444444444444	44444444

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	19(101^15)	=	220584101520	299721681721	64978519					1	47(101^15)	=	545655409023	899311528469	34420547		1	76(101^15)	=	882336406081	198886726886	59914076
2	14(101^14)	=	1609263898	532671236865	O8759614					2	86(101^14)	=	9885478233	843551883599	82380486		2	57(101^14)	=	6552003015	454447178664	99949857
3	25(101^13)	=	28452332	O10832235446	69532525					3	12(101^13)	=	13657119	365199473014	41375612		3	42(101^12)	=	473266	512655427282	67770442
4	35(101^12)	=	394388	760546189402	23142035					4	89(101^12)	=	1002874	276817453051	38846889		4	58(101^11)	=	6470	876410658836	O1963858
5	90(101^11)	=	10041	O15119987848	99599090					5	74(101^11)	=	8255	945765323342	50781474		5	49(101^10)	=	54	126484145149	O2099049
6	37(101^10)	=	40	871018640214	56687037					6	43(101^10)	=	47	498751392681	79393043		6	65(101^9)	=		710895427244	83458565
7	66(101^9)	=		721832279971	67819466					7	66(101^9)	=		721832279971	67819466		7	84(101^8)	=		9095996327	27587284
8	96(101^8)	=		10395424374	O2956896					8	39(101^8)	=		4223141151	94951239		8	12(101^7)	=		12865624	22528412
9	78(101^7)	=		83626557	46434678					9	96(101^7)	=		102924993	80227296		9	43(101^6)	=		456453	66475843
10	86(101^6)	=		912907	32951686					10	14(101^6)	=		148612	82108414		10	61(101^5)	=		6411	16130561
11	65(101^5)	=		6831	56532565					11	63(101^5)	=		6621	36331563		11	8(101^4)	=		8	32483208
12	77(101^4)	=		80	12650877					12	42(101^4)	=		43	70536842		12	8(101^3)	=			8242408
13	77(101^3)	=			79333177					13	93(101^3)	=			95817993		13	27(101^2)	=			275427
14	82(101^2)	=			836482					14	55(101^2)	=			561055		14	98(101^1)	=			9898
15	75(101^1)	=			7575					15	36(101^1)	=			3636					888888888886	2888888888884	484825981
222222222219	3196269998922	694153912								555555555551	4555555555547	855555555					888888888888	888888888888	88888888
222222222222	222222222222	22222222								555555555555	555555555555	55555555

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	28(101^15)	=	325071307503	599589846747	69442028					1	57(101^15)	=	661752304560	899165045164	94935557		1	86(101^15)	=	998433301618	198740243582	20429086
2	71(101^14)	=	8161266913	987118415530	O8709471					2	42(101^14)	=	4827791695	598013710595	26278842		2	13(101^14)	=	1494316477	208909005660	43848213
3	88(101^13)	=	100152208	678129468772	36754488					3	76(101^13)	=	86495089	312929995757	95378876		3	63(101^13)	=	71699876	667297233325	67221963
4	53(101^12)	=	597217	265969943951	95043653					4	6(101^12)	=	67609	501807918183	23967206		4	60(101^12)	=	676095	O18079181832	39672060
5	85(101^11)	=	9483	180946655190	71843585					5	69(101^11)	=	7698	111591990684	23025969		5	53(101^11)	=	5913	O42237326177	74208353
6	5(101^10)	=	5	523110627056	O2255005					6	11(101^10)	=	12	150843379523	24961011		6	17(101^10)	=	18	778576131990	47667017
7	100(101^9)	=	1	O93685272684	36090100					7	99(101^9)	=	1	O82748419957	51729199		7	99(101^9)	=	1	O82748419957	51729199
8	44(101^8)	=		4764569504	76355244					8	88(101^8)	=		9529139009	52710488		8	31(101^8)	=		3356855787	44704831
9	17(101^7)	=		18226300	98581917					9	34(101^7)	=		36452601	97163834		9	51(101^7)	=		54678902	95745751
10	28(101^6)	=		297225	64216828					10	57(101^6)	=		605066	48584257		10	86(101^6)	=		912907	32951686
11	98(101^5)	=		10299	89849098					11	96(101^5)	=		10089	69648096		11	93(101^5)	=		9774	39346593
12	66(101^4)	=		68	67986466					12	31(101^4)	=		32	25872431		12	97(101^4)	=		100	93858897
13	15(101^3)	=			15454515					13	31(101^3)	=			31939331		13	47(101^3)	=			48424147
14	73(101^2)	=			744673					14	46(101^2)	=			469246		14	18(101^2)	=			183618
15	62(101^1)	=			6262					15	23(101^1)	=			2323		15	85(101^1)				8585
333333333330	3239648060642	722368857								666666666664	2583918246703	666666666					999999999998	1856935072027	699999999
333333333333	333333333333	33333333								666666666666	666666666666	66666666					999999999999	999999999999	99999999

-	B.S.Rangaswamy

S U M   O F   M U L T I P L E   P O W E R S   O F   " 1 0 1 "

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	9(101^15)	=	104487205983	299868165026	O4463509					1	38(101^15)	=	441168203040	599443363443	29957038		1	66(101^15)	=	766239510544	199033210190	99399066
2	57(101^14)	=	6552003015	454447178664	99949857					2	28(101^14)	=	3218527797	O65342473730	17519228		2	100(101^14)	=	11494742132	376223120464	91140100
3	63(101^13)	=	71699876	667297233325	67221963					3	50(101^13)	=	56904664	O21664470893	39065050		3	38(101^13)	=	43247544	656464997878	97689438
4	17(101^12)	=	191560	255122434852	51240417					4	71(101^12)	=	800045	771393698501	66945271		4	24(101^12)	=	270438	O07231672732	95868824
5	95(101^11)	=	10598	849293320507	27354595					5	79(101^11)	=	8813	779938656000	78536979		5	63(101^11)	=	7028	710583991494	29719363
6	69(101^10)	=	76	218926653373	11119069					6	75(101^10)	=	82	846659405840	33825075		6	81(101^10)	=	89	474392158307	56531081
7	33(101^9)	=		360916139985	83909733					7	32(101^9)	=		349979287258	99548832		7	32(101^9)	=		349979287258	99548832
8	48(101^8)	=		5197712187	O1478448					8	92(101^8)	=		9962281691	77833692		8	35(101^8)	=		3789998469	69828035
9	39(101^7)	=		41813278	73217339					9	56(101^7)	=		60039579	71799256		9	74(101^7)	=		79338016	O5591874
10	43(101^6)	=		456453	66475843					10	72(101^6)	=		764294	50843272		10	28(101^5)	=		2942	82814028
11	32(101^5)	=		3363	23216032					11	30(101^5)	=		3153	O3015030		11	19(101^4)	=		19	77147619
12	89(101^4)	=		92	61375689					12	54(101^4)	=		56	19261654		12	70(101^3)	=			72121070
13	38(101^3)	=			39151438					13	54(101^3)	=			55636254		13	37(101^2)	=			377437
14	91(101^2)	=			928291					14	64(101^2)	=			652864		14	10(101^1)	=			1010
15	88(101^1)	=			8888					15	49(101^1)	=			4949					777777777775	2770546105037	872185903
111111111108	3111111111105	605169154								444444444441	3357977499815	641429414					777777777777	777777777777	77777777
111111111111	111111111111	11111111								444444444444	444444444444	44444444

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	19(101^15)	=	220584101520	299721681721	64978519					1	47(101^15)	=	545655409023	899311528469	34420547		1	76(101^15)	=	882336406081	198886726886	59914076
2	14(101^14)	=	1609263898	532671236865	O8759614					2	86(101^14)	=	9885478233	843551883599	82380486		2	57(101^14)	=	6552003015	454447178664	99949857
3	25(101^13)	=	28452332	O10832235446	69532525					3	12(101^13)	=	13657119	365199473014	41375612		3	42(101^12)	=	473266	512655427282	67770442
4	35(101^12)	=	394388	760546189402	23142035					4	89(101^12)	=	1002874	276817453051	38846889		4	58(101^11)	=	6470	876410658836	O1963858
5	90(101^11)	=	10041	O15119987848	99599090					5	74(101^11)	=	8255	945765323342	50781474		5	49(101^10)	=	54	126484145149	O2099049
6	37(101^10)	=	40	871018640214	56687037					6	43(101^10)	=	47	498751392681	79393043		6	65(101^9)	=		710895427244	83458565
7	66(101^9)	=		721832279971	67819466					7	66(101^9)	=		721832279971	67819466		7	84(101^8)	=		9095996327	27587284
8	96(101^8)	=		10395424374	O2956896					8	39(101^8)	=		4223141151	94951239		8	12(101^7)	=		12865624	22528412
9	78(101^7)	=		83626557	46434678					9	96(101^7)	=		102924993	80227296		9	43(101^6)	=		456453	66475843
10	86(101^6)	=		912907	32951686					10	14(101^6)	=		148612	82108414		10	61(101^5)	=		6411	16130561
11	65(101^5)	=		6831	56532565					11	63(101^5)	=		6621	36331563		11	8(101^4)	=		8	32483208
12	77(101^4)	=		80	12650877					12	42(101^4)	=		43	70536842		12	8(101^3)	=			8242408
13	77(101^3)	=			79333177					13	93(101^3)	=			95817993		13	27(101^2)	=			275427
14	82(101^2)	=			836482					14	55(101^2)	=			561055		14	98(101^1)	=			9898
15	75(101^1)	=			7575					15	36(101^1)	=			3636					888888888886	2888888888884	484825981
222222222219	3196269998922	694153912								555555555551	4555555555547	855555555					888888888888	888888888888	88888888
222222222222	222222222222	22222222								555555555555	555555555555	55555555

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	28(101^15)	=	325071307503	599589846747	69442028					1	57(101^15)	=	661752304560	899165045164	94935557		1	86(101^15)	=	998433301618	198740243582	20429086
2	71(101^14)	=	8161266913	987118415530	O8709471					2	42(101^14)	=	4827791695	598013710595	26278842		2	13(101^14)	=	1494316477	208909005660	43848213
3	88(101^13)	=	100152208	678129468772	36754488					3	76(101^13)	=	86495089	312929995757	95378876		3	63(101^13)	=	71699876	667297233325	67221963
4	53(101^12)	=	597217	265969943951	95043653					4	6(101^12)	=	67609	501807918183	23967206		4	60(101^12)	=	676095	O18079181832	39672060
5	85(101^11)	=	9483	180946655190	71843585					5	69(101^11)	=	7698	111591990684	23025969		5	53(101^11)	=	5913	O42237326177	74208353
6	5(101^10)	=	5	523110627056	O2255005					6	11(101^10)	=	12	150843379523	24961011		6	17(101^10)	=	18	778576131990	47667017
7	100(101^9)	=	1	O93685272684	36090100					7	99(101^9)	=	1	O82748419957	51729199		7	99(101^9)	=	1	O82748419957	51729199
8	44(101^8)	=		4764569504	76355244					8	88(101^8)	=		9529139009	52710488		8	31(101^8)	=		3356855787	44704831
9	17(101^7)	=		18226300	98581917					9	34(101^7)	=		36452601	97163834		9	51(101^7)	=		54678902	95745751
10	28(101^6)	=		297225	64216828					10	57(101^6)	=		605066	48584257		10	86(101^6)	=		912907	32951686
11	98(101^5)	=		10299	89849098					11	96(101^5)	=		10089	69648096		11	93(101^5)	=		9774	39346593
12	66(101^4)	=		68	67986466					12	31(101^4)	=		32	25872431		12	97(101^4)	=		100	93858897
13	15(101^3)	=			15454515					13	31(101^3)	=			31939331		13	47(101^3)	=			48424147
14	73(101^2)	=			744673					14	46(101^2)	=			469246		14	18(101^2)	=			183618
15	62(101^1)	=			6262					15	23(101^1)	=			2323		15	85(101^1)	=			8585
333333333330	3239648060642	722368857								666666666664	2583918246703	666666666					999999999998	1856935072027	699999999
333333333333	333333333333	33333333								666666666666	666666666666	66666666					999999999999	999999999999	99999999

-	B.S.Rangaswamy

Sum of multiple powers of palindrome prime "353"									            32  digit  random  palindrome							      Sum of multiple powers of "11" -  32 digit random palindrome

SN	Multi power		            E X P A N D E D   P O W E R S				SN	          Squares			            E X P A N D E D   P O W E R S					SN	Multiple power		            E X P A N D E D   P O W E R S
1	2(353^12)	=	7487321	129479627316	927293927682	A	1	616324	7639200445^2	=	37985621	462109858667	O38888198025		C	1	2(11^30)	=	34898804	537772814637	117607507602
2	341(353^11)	=	3616397	316080103279	139103724277		2		90681241^2	=		8223	O87469300081			2	(11^29)	=	1586309	297171491574	414436704891
3	246(353^10)	=	7390	641919331622	595334664054		3		7427^2	=			55160329			3	10(11^28)	=	1442099	361064992340	376760640810
4	23(353^9)	=	1	957492850418	334366207159		4		23^2	=			529			4	4(11^27)	=	52439	976765999721	468245841484
5	274(353^8)	=		66061465822	715065444144		5		3^2	=			9			5	5(11^26)	=	5959	O88268863604	712300663805
6	113(353^7)	=		77179397	O13779723281						37985621	462109866890	126412658973			6	10(11^23)	=	8	954302432552	373722465310
7	285(353^6)	=		551433	431495580765						37985621	462109866890	126412658973			7	3(11^22)	=		244208248160	519283339963
8	332(353^5)	=		1819	749508041676											8	3(11^20)	=		2018249984	797680027603
9	77(353^4)	=		1	195610021837		SN	    Squares & Cubes			            E X P A N D E D   P O W E R S					9	8(11^19)	=		489272723	587316370328
10	217(353^3)	=			9545174009	B	1	3	3615513156^3	=	37985621	459853960535	381468900416			10	8(11^18)	=		44479338	507937851848
11	69(353^2)	=			8598021		2	4	7496382543^2	=		2255906354	670995146849			11	5(11^17)	=		2527235	142496468855
12	12(353^1)	=			4236		3		271934^2	=			73948100356			12	10(11^16)	=		459497	298635721610
11111109	2111111111107	4097331387860		4		715^2	=			511225			13	8(11^15)	=		33417	985355325208
11111111	111111111111	111111111111		5		10^2	=			100			14	5(11^14)	=		1898	749167916205
6		3^3	=			27			15	5(11^13)	=		172	613560719655
37985621	462109866889	1126412658973			16	9(11^12)	=		28	245855390489
On similar lines, the other EIGHT repdigital palindromes of										37985621	462109866890	126412658973			17	5(11^11)	=		1	426558353055
18	8(11^10)	=			207499396808
32 digits can be generated from multiple powers of " 353 "															19	4(11^9)	=			9431790764
The 32 digit random palindrome is expressed as sum of								20	9(11^8)	=			1929229929
a palindrome prime.															21	4(11^6)	=			7086244
A.	Squares							22	3(11^5)	=			483153
23	(11^4)	=			14641
B.	Squares and Cubes							24	5(11^3)	=			6655
25	6(11^2)	=			726
C.	Multiple powers of "11" the lowest palindrome prime.							26	10(11^1)	=			110
37985618	3373841003277	8555789317751
37985621	462109866890	126412658973

3rd  September 2009
Dear Mr Patrick

Hope the temperature is coming down appreciably and you are getting into good health and spirits.

Please find herewith enclosed all repdigital palindromes of 32 digits, generated by multiple powers of palindrome prime 353 at Excel sheet 2 of
enclosure for your kind perusal. Next to PP 101, this is the only palindrome
prime in 3, 4 and 5 digits, which is capable of generating repdigitals of 32 digits.
By coincidence 353 also happens to be my house number.

Also enclosed at sheet 3, the random palindrome of 32 digits, generated by

A.	Squares
B.	Squares and Cubes
C.	Multiple powers of second lowest palindrome prime 11
D.	Multiple powers of prime 5503

Next to 11, 5503 is the only number in 2, 3 and 4 digits, which is capable of generating this particular Random palindrome of 32 digits!

This concludes my submissions to you on the subject  Repdigitals & Random palindrome as Sum of multiple powers of primes.

Thanking You

With Regards

B.S.Rangaswamy

S U M   O F   M U L T I P L E   P O W E R S   O F   " 1 0 1 "

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	9(101^15)	=	104487205983	299868165026	O4463509					1	38(101^15)	=	441168203040	599443363443	29957038		1	66(101^15)	=	766239510544	199033210190	99399066
2	57(101^14)	=	6552003015	454447178664	99949857					2	28(101^14)	=	3218527797	O65342473730	17519228		2	100(101^14)	=	11494742132	376223120464	91140100
3	63(101^13)	=	71699876	667297233325	67221963					3	50(101^13)	=	56904664	O21664470893	39065050		3	38(101^13)	=	43247544	656464997878	97689438
4	17(101^12)	=	191560	255122434852	51240417					4	71(101^12)	=	800045	771393698501	66945271		4	24(101^12)	=	270438	O07231672732	95868824
5	95(101^11)	=	10598	849293320507	27354595					5	79(101^11)	=	8813	779938656000	78536979		5	63(101^11)	=	7028	710583991494	29719363
6	69(101^10)	=	76	218926653373	11119069					6	75(101^10)	=	82	846659405840	33825075		6	81(101^10)	=	89	474392158307	56531081
7	33(101^9)	=		360916139985	83909733					7	32(101^9)	=		349979287258	99548832		7	32(101^9)	=		349979287258	99548832
8	48(101^8)	=		5197712187	O1478448					8	92(101^8)	=		9962281691	77833692		8	35(101^8)	=		3789998469	69828035
9	39(101^7)	=		41813278	73217339					9	56(101^7)	=		60039579	71799256		9	74(101^7)	=		79338016	O5591874
10	43(101^6)	=		456453	66475843					10	72(101^6)	=		764294	50843272		10	28(101^5)	=		2942	82814028
11	32(101^5)	=		3363	23216032					11	30(101^5)	=		3153	O3015030		11	19(101^4)	=		19	77147619
12	89(101^4)	=		92	61375689					12	54(101^4)	=		56	19261654		12	70(101^3)	=			72121070
13	38(101^3)	=			39151438					13	54(101^3)	=			55636254		13	37(101^2)	=			377437
14	91(101^2)	=			928291					14	64(101^2)	=			652864		14	10(101^1)	=			1010
15	88(101^1)	=			8888					15	49(101^1)	=			4949					777777777775	2770546105037	872185903
111111111108	3111111111105	605169154								444444444441	3357437499815	641429414					777777777777	777777777777	77777777
111111111111	111111111111	11111111								444444444444	444444444444	44444444

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	19(101^15)	=	220584101520	299721681721	64978519					1	47(101^15)	=	545655409023	899311528469	34420547		1	76(101^15)	=	882336406081	198886726886	59914076
2	14(101^14)	=	1609263898	532671236865	O8759614					2	86(101^14)	=	9885478233	843551883599	82380486		2	57(101^14)	=	6552003015	454447178664	99949857
3	25(101^13)	=	28452332	O10832235446	69532525					3	12(101^13)	=	13657119	365199473014	41375612		3	42(101^12)	=	473266	512655427282	67770442
4	35(101^12)	=	394388	760546189402	23142035					4	89(101^12)	=	1002874	276817453051	38846889		4	58(101^11)	=	6470	876410658836	O1963858
5	90(101^11)	=	10041	O15119987848	99599090					5	74(101^11)	=	8255	945765323342	50781474		5	49(101^10)	=	54	126484145149	O2099049
6	37(101^10)	=	40	871018640214	56687037					6	43(101^10)	=	47	498751392681	79393043		6	65(101^9)	=		710895427244	83458565
7	66(101^9)	=		721832279971	67819466					7	66(101^9)	=		721832279971	67819466		7	84(101^8)	=		9095996327	27587284
8	96(101^8)	=		10395424374	O2956896					8	39(101^8)	=		4223141151	94951239		8	12(101^7)	=		12865624	22528412
9	78(101^7)	=		83626557	46434678					9	96(101^7)	=		102924993	80227296		9	43(101^6)	=		456453	66475843
10	86(101^6)	=		912907	32951686					10	14(101^6)	=		148612	82108414		10	61(101^5)	=		6411	16130561
11	65(101^5)	=		6831	56532565					11	63(101^5)	=		6621	36331563		11	8(101^4)	=		8	32483208
12	77(101^4)	=		80	12650877					12	42(101^4)	=		43	70536842		12	8(101^3)	=			8242408
13	77(101^3)	=			79333177					13	93(101^3)	=			95817993		13	27(101^2)	=			275427
14	82(101^2)	=			836482					14	55(101^2)	=			561055		14	98(101^1)	=			9898
15	75(101^1)	=			7575					15	36(101^1)	=			3636					888888888886	2888888888884	484825981
222222222219	3196269998922	694153912								555555555551	4555555555547	855555555					888888888888	888888888888	88888888
222222222222	222222222222	22222222								555555555555	555555555555	55555555

SN	Multi power		            E X P A N D E D   P O W E R S							SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	28(101^15)	=	325071307503	599589846747	69442028					1	57(101^15)	=	661752304560	899165045164	94935557		1	86(101^15)	=	998433301618	198740243582	20429086
2	71(101^14)	=	8161266913	987118415530	O8709471					2	42(101^14)	=	4827791695	598013710595	26278842		2	13(101^14)	=	1494316477	208909005660	43848213
3	88(101^13)	=	100152208	678129468772	36754488					3	76(101^13)	=	86495089	312929995757	95378876		3	63(101^13)	=	71699876	667297233325	67221963
4	53(101^12)	=	597217	265969943951	95043653					4	6(101^12)	=	67609	501807918183	23967206		4	60(101^12)	=	676095	O18079181832	39672060
5	85(101^11)	=	9483	180946655190	71843585					5	69(101^11)	=	7698	111591990684	23025969		5	53(101^11)	=	5913	O42237326177	74208353
6	5(101^10)	=	5	523110627056	O2255005					6	11(101^10)	=	12	150843379523	24961011		6	17(101^10)	=	18	778576131990	47667017
7	100(101^9)	=	1	O93685272684	36090100					7	99(101^9)	=	1	O82748419957	51729199		7	99(101^9)	=	1	O82748419957	51729199
8	44(101^8)	=		4764569504	76355244					8	88(101^8)	=		9529139009	52710488		8	31(101^8)	=		3356855787	44704831
9	17(101^7)	=		18226300	98581917					9	34(101^7)	=		36452601	97163834		9	51(101^7)	=		54678902	95745751
10	28(101^6)	=		297225	64216828					10	57(101^6)	=		605066	48584257		10	86(101^6)	=		912907	32951686
11	98(101^5)	=		10299	89849098					11	96(101^5)	=		10089	69648096		11	93(101^5)	=		9774	39346593
12	66(101^4)	=		68	67986466					12	31(101^4)	=		32	25872431		12	97(101^4)	=		100	93858897
13	15(101^3)	=			15454515					13	31(101^3)	=			31939331		13	47(101^3)	=			48424147
14	73(101^2)	=			744673					14	46(101^2)	=			469246		14	18(101^2)	=			183618
15	62(101^1)	=			6262					15	23(101^1)	=			2323		15	85(101^1)				8585
333333333330	3239648060642	722368857								666666666664	2583918246703	666666666					999999999998	1856935072027	699999999
333333333333	333333333333	33333333								666666666666	666666666666	66666666					999999999999	999999999999	99999999

-	B.S.Rangaswamy

Repdigital palindromes as sum of multiple powers of pal prime "353"

SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	2(353^12)	=	7487321	129479627316	927293927682		1	11(353^12)	=	41180266	212137950243	100116602251		1	20(353^12)	=	74873211	294796273169	272939276820
2	341(353^11)	=	3616397	316080103279	139103724277		2	307(353^11)	=	3255818	111544257204	386231212179		2	273(353^11)	=	2895238	907008411129	633358700081
3	246(353^10)	=	7390	641919331622	595334664054		3	278(353^10)	=	8352	O26234041427	160581449622		3	310(353^10)	=	9313	410548751231	725828235190
4	23(353^9)	=	1	957492850418	334366207159		4	95(353^9)	=	8	O85296556075	728903899135		4	166(353^9)	=	14	127991876932	326295234278
5	274(353^8)	=		66061465822	715065444144		5	38(353^8)	=		9161809128	697709806118		5	155(353^8)	=		37370537235	477500524955
6	113(353^7)	=		77179397	O13779723281		6	102(353^7)	=		69666358	366420635174		6	90(353^7)	=		61470316	205665266330
7	285(353^6)	=		551433	431495580765		7	84(353^6)	=		162527	748230276436		7	236(353^6)	=		456625	578361252844
8	332(353^5)	=		1819	749508041676		8	269(353^5)	=		1474	435595371117		8	207(353^5)	=		1134	602855917551
9	77(353^4)	=		1	195610021837		9	310(353^4)	=		4	813494893110		9	190(353^4)	=		2	950206547390
10	217(353^3)	=			9545174009		10	162(353^3)	=			7125890274		10	108(353^3)	=			4750593516
11	69(353^2)	=			8598021		11	276(353^2)	=			34392084		11	130(353^2)	=			16199170
12	12(353^1)	=			4236		12	48(353^1)	=			16944		12	84(353^1)	=			29652
11111109	2111111111107	4097331387860					44444444	332913846938	4444444444444					77777776	1777777777773	4777777777777
11111111	111111111111	111111111111					44444444	444444444444	444444444444					77777777	777777777777	777777777777

SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	5(353^12)	=	18718302	823699068292	318234819205		1	14(353^12)	=	52411247	906357391218	491057493774		1	23(353^12)	=	86104192	989015714144	663880168343
2	330(353^11)	=	3499739	338142035431	424939088010		2	296(353^11)	=	3139160	133606189356	672066575912		2	262(353^11)	=	2778580	929070343281	919194063814
3	139(353^10)	=	4176	O13117020713	580290724811		3	171(353^10)	=	5137	397431730518	145537510379		3	203(353^10)	=	6098	781746440322	710784295947
4	47(353^9)	=	4	O00094085637	465878771151		4	118(353^9)	=	10	O42789406494	O63270106294		4	190(353^9)	=	16	170593112151	457807798270
5	195(353^8)	=		47014546844	632984531395		5	312(353^8)	=		75223274951	412775250232		5	76(353^8)	=		18323618257	395419612236
6	227(353^7)	=		155041797	540955727299		6	216(353^7)	=		147528758	893596639192		6	204(353^7)	=		139332716	732841270348
7	218(353^6)	=		421798	203740479322		7	17(353^6)	=		32892	520475174993		7	169(353^6)	=		326990	350606151401
8	311(353^5)	=		1704	644870484823		8	249(353^5)	=		1364	812131031257		8	186(353^5)	=		1019	498218360698
9	155(353^4)	=		2	406747446555		9	35(353^4)	=			543459100835		9	267(353^4)	=		4	145816569227
10	81(353^3)	=			3562945137		10	26(353^3)	=			1143661402		10	325(353^3)	=			14295767525
11	138(353^2)	=			17196042		11	345(353^2)	=			42990105		11	199(353^2)	=			24797191
12	24(353^1)	=			8472		12	60(353^1)	=			21180		12	96(353^1)	=			33888
22222221	1209011115868	4222222222222					55555554	1512766149057	4492285449261					88888886	2888888888884	4888888888888
22222222	222222222222	222222222222					55555555	555555555555	555555555555					88888888	888888888888	888888888888

SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S				SN	Multi power		            E X P A N D E D   P O W E R S
1	8(353^12)	=	29949284	517918509267	709175710728		1	17(353^12)	=	63642229	600576832193	881998385297		1	26(353^12)	=	97335174	683235155120	O54821059866
2	319(353^11)	=	3383081	360203967583	710774451743		2	285(353^11)	=	3022502	155668121508	957901939645		2	251(353^11)	=	2661922	951132275434	205029427547
3	32(353^10)	=	961	384314709804	565246785568		3	64(353^10)	=	1922	768629419609	130493571136		3	96(353^10)	=	2884	152944129413	695740356704
4	71(353^9)	=	6	O42695320856	597391335143		4	142(353^9)	=	12	O85390641713	194782670286		4	213(353^9)	=	18	128085962569	792174005429
5	116(353^8)	=		27967627866	550903618676		5	233(353^8)	=		56176355973	330694337513		5	350(353^8)	=		84385084080	110485056350
6	341(353^7)	=		232904198	O68131731317		6	329(353^7)	=		224708155	907376362473		6	318(353^7)	=		217195117	260017274366
7	151(353^6)	=		292162	975985377879		7	303(353^6)	=		586260	806116354287		7	102(353^6)	=		197355	122851049958
8	290(353^5)	=		1589	540232927970		8	228(353^5)	=		1249	707493474404		8	165(353^5)	=		904	393580803845
9	232(353^4)	=		3	602357468392		9	112(353^4)	=		1	739069122672		9	345(353^4)	=		5	356953993945
10	298(353^3)	=			13108119146		10	244(353^3)	=			10732822388		10	189(353^3)	=			8313538653
11	207(353^2)	=			25794063		11	61(353^2)	=			7601149		11	268(353^2)	=			33395212
12	36(353^1)	=			12708		12	72(353^1)	=			25416		12	108(353^1)	=			38124
33333332	1290638012472	5265201602016					66666665	1581276024948	5666666666666					99999998	1999999999997	2945178940133
33333333	333333333333	333333333333					66666666	666666666666	666666666666					99999999	999999999999	999999999999

-	B.S.Rangaswamy

32  digit  random  palindrome							      Sum of multiple powers of "11" -  32 digit random palindrome							   Sum of multiple powers of prime "5503"  -  Random palindrome

SN	          Squares			            E X P A N D E D   P O W E R S					SN	Multiple power		            E X P A N D E D   P O W E R S				SN	Multiple power		            E X P A N D E D   P O W E R S
A	1	616324	7639200445^2	=	37985621	462109858667	O38888198025		C	1	2(11^30)	=	34898804	537772814637	117607507602		1	45(5503^8)	=	37845009	297226706356	O60165555245
2		90681241^2	=		8223	O87469300081			2	(11^29)	=	1586309	297171491574	414436704891		2	920(5503^7)	=	140599	707446235668	817313192040
3		7427^2	=			55160329			3	10(11^28)	=	1442099	361064992340	376760640810		3	448(5503^6)	=	12	441567235245	909272838592
4		23^2	=			529			4	4(11^27)	=	52439	976765999721	468245841484		4	3144(5503^5)	=		15866462200	655624023992
5		3^2	=			9			5	5(11^26)	=	5959	O88268863604	712300663805		5	3519(5503^4)	=		3227136	371377071039
37985621	462109866890	126412658973			6	10(11^23)	=	8	954302432552	373722465310		6	1694(5503^3)	=		282	300693104738
37985621	462109866890	126412658973			7	3(11^22)	=		244208248160	519283339963		7	395(5503^2)	=			11961788555
8	3(11^20)	=		2018249984	797680027603		8	924(5503^1)	=			5084772
SN	    Squares & Cubes			            E X P A N D E D   P O W E R S					9	8(11^19)	=		489272723	587316370328					37985620	1462109866887	3066247103728
B	1	3	3615513156^3	=	37985621	459853960535	381468900416			10	8(11^18)	=		44479338	507937851848					37985621	462109866890	126412658973
2	4	7496382543^2	=		2255906354	670995146849			11	5(11^17)	=		2527235	142496468855
3		271934^2	=			73948100356			12	10(11^16)	=		459497	298635721610
4		715^2	=			511225			13	8(11^15)	=		33417	985355325208
5		10^2	=			100			14	5(11^14)	=		1898	749167916205
6		3^3	=			27			15	5(11^13)	=		172	613560719655
37985621	462109866889	1126412658973			16	9(11^12)	=		28	245855390489
37985621	462109866890	126412658973			17	5(11^11)	=		1	426558353055
18	8(11^10)	=			207499396808
19	4(11^9)	=			9431790764
The 32 digit random palindrome is expressed as sum of								20	9(11^8)	=			1929229929
21	4(11^6)	=			7086244
A.	Squares							22	3(11^5)	=			483153
23	(11^4)	=			14641
B.	Squares and Cubes							24	5(11^3)	=			6655
25	6(11^2)	=			726
C.	Multiple powers of "11" the lowest palindrome prime.							26	10(11^1)	=			110
37985618	3373841003277	8555789317751
D.	Multiple powers of prime " 5503 ".										37985621	462109866890	126412658973

-	B.S.Rangaswamy

Dt 18th September 2009
Dear Mr Patrick

SUB: Palindrome as sum of Powers  WON 178

63 digit palindrome a dream sum of different powers as prescribed in your WON Plate 178, has now become a reality! This Palindrome is:

34336838202925124846578490892810182980948756484215292028386334

which is the sum of 21 different powers, the degree of powers ranging from 2 to 70 and is depicted in the following list:

--------------------------------------------------------------
1.    90^32                                    12.     7^17
2.    48^18                                    13.    93^7
3.    14^23                                    14.    67^6
4.    19^20                                    15.     5^13
5.     3^50                                    16.     2^27
6.    33^15                                    17.   256^3
7.     2^70                                    18.    13^4
8.   121^10                                    19.     2^12
9.     2^67                                    20.    43^2
10.   272^8                                     21.     2^5
11.    79^9
---------------------------------------------------------------

There also exists a higher value 63 digit palindrome, which happens to be the sum of 22 different powers!! These are not sporadic, but engineered by an Engineer.

I shall be submitting details of both these sums in Excel format, when complete for your perusal. Kindly bear with this requisite and reasonable delay.

Thanking You

With High Regards

B.S.Rangaswamy

Dear Mr Patrick

Extremely sorry for missing '3', which is now included at the end of 63 digit palindrome.

- B.S.Rangaswamy

Dt 18th September 2009
Dear Mr Patrick

SUB: Palindrome as sum of Powers  WON 178

63 digit palindrome a dream sum of different powers as prescribed in your WON Plate 178, has now become a reality! This Palindrome is:

343368382029251248465784908928101829809487564842152920283863343

which is the sum of 21 different powers, the degree of powers ranging from 2 to 70 and is depicted in the following list:

--------------------------------------------------------------
1.    90^32                                    12.     7^17
2.    48^18                                    13.    93^7
3.    14^23                                    14.    67^6
4.    19^20                                    15.     5^13
5.     3^50                                    16.     2^27
6.    33^15                                    17.   256^3
7.     2^70                                    18.    13^4
8.   121^10                                    19.     2^12
9.     2^67                                    20.    43^2
10.   272^8                                     21.     2^5
11.    79^9
---------------------------------------------------------------

There also exists a higher value 63 digit palindrome, which happens to be the sum of 22 different powers!! These are not sporadic, but engineered by an Engineer.

I shall be submitting details of both these sums in Excel format, when complete for your perusal. Kindly bear with this requisite and reasonable delay.

Thanking You

With High Regards

B.S.Rangaswamy

Dt 22nd September 2009
Dear Mr Patrick

SUB: Palindromes as sum of Powers - WON 178

Further to my submissions dated 18th September, the second 63 digit palindrome is:

343368382029251248465784908928111829809487564842152920283863343
( Sheet 2 of Excel format; Earlier palindrome at Sheet 1 )

which is the sum of 22 different powers, and is illustrated in the following list
with only the addition of 10^31 to the first:

-------------------------------------------------------------
1.    90^32                                   12.    79^9
2.    10^31                                   13.     7^17
3.    48^18                                   14.    93^7
4.    14^23                                   15.    67^6
5.    19^20                                   16.     5^13
6.     3^50                                   17.     2^27
7.    33^15                                   18.   256^3
8.     2^70                                   19.    13^4
9.   121^10                                   20.     2^12
10.     2^67                                   21.    43^2
11.   272^8                                    22.     2^5

---------------------------------------------------------------

Various constituent powers for the following palindromes may be worked out by enthusiastic readers:

343368382029251248465784908928121829809487564842152920283863343

343368382029251248465784908928131829809487564842152920283863343

3433 - - - - - - - - - - - - - - - - - - - - - - 141 - - - - - - - - - - - - - - - - - - - - - -3343

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

343368382029251248465784908928191829809487564842152920283863343

Arriving at the constituent powers for the following 63 digit palindrome
can turn out to be highly daunting / haunting task to the involved reader and likewise for the next lower values of 63 digit palindromes:

343368382029251248465784908928090829809487564842152920283863343

Grandsons easy and harmonic way of calculating high degree powers was of great help. He is now nearer to you in England for higher studies. It was a daunting experience to search and get at the last few appropriate powers to arrive at the first 63 digit palindrome.

Thank You immensely for bestowing me an opportunity to participate in this highly interesting and educating topic.

With High Regards

B.S.Rangaswamy

63 Digit Palindrome - Sum of Powers										         63 Digit Palindrome - Sum of Powers

A								      POWERS			A
SN	Powers	                                   E   X   P   A   N   D   E   D       P   O   W   E   R   S					ORDER	 SORT		SN	Powers	                                   E   X   P   A   N   D   E   D       P   O   W   E   R   S
1	90^32	343368382029251	248465784908	928100000000	O00000000000	O00000000000	32	70		1	90^32	343368382029251	248465784908	928100000000	O00000000000	O00000000000
2	48^18			1829541	532030568071	946613817344	18	67		2	48^18			1829541	532030568071	946613817344
3	14^23			229	585692886981	495482220544	23	50		3	14^23			229	585692886981	495482220544
4	19^20			37	589973457545	958193355601	20	32		4	19^20			37	589973457545	958193355601
5	3^50				717897987691	852588770249	50	27		5	3^50				717897987691	852588770249
6	33^15				59938945498	865420543457	15	23		6	33^15				59938945498	865420543457
7	2^70				1180591620	717411303424	70	20		7	2^70				1180591620	717411303424
8	121^10				672749994	932560009201	10	18		8	121^10				672749994	932560009201
9	2^67				147573952	589676412928	67	17		9	2^67				147573952	589676412928
10	272^8				29960650	O73923649536	8	15		10	272^8				29960650	O73923649536
11	79^9				119851	595982618319	9	13		11	79^9				119851	595982618319
12	7^17				232	630513987207	17	12		12	7^17				232	630513987207
13	93^7				60	170087060757	7	10		13	93^7				60	170087060757
14	67^6					90458382169	6	9		14	67^6					90458382169
15	5^13					1220703125	13	8		15	5^13					1220703125
16	2^27					134217728	27	7		16	2^27					134217728
17	256^3					16777216	3	6		17	256^3					16777216
18	13^4					28561	4	5		18	13^4					28561
19	2^12					4096	12	4		19	2^12					4096
20	43^2					1849	2	3		20	43^2					1849
21	2^5					32	5	2		21	2^5					32
343368382029251	248465784908	928101829807	2487564842145	7920283863343						343368382029251	248465784908	928101829807	2487564842145	7920283863343	  < Total of individual columns
343368382029251	248465784908	928101829809	487564842152	920283863343	< 63 Digit Palindrome					343368382029251	248465784908	928101829809	487564842152	920283863343	< 63 Digit Palindrome

-	B.S.Rangaswamy
21.09.2009

63 Digit Palindrome - Sum of Powers													         63 Digit Palindrome - Sum of Powers

B								      POWERS					B
SN	Powers	                                   E   X   P   A   N   D   E   D       P   O   W   E   R   S					ORDER	 SORT					SN	Powers	                                   E   X   P   A   N   D   E   D       P   O   W   E   R   S
1	90^32	343368382029251	248465784908	928100000000	O00000000000	O00000000000	32	70					1	90^32	343368382029251	248465784908	928100000000	O00000000000	O00000000000
Add	2	10^31			10000000	O00000000000	O00000000000	31	67				Add	2	10^31			10000000	O00000000000	O00000000000
to A	3	48^18			1829541	532030568071	946613817344	18	50				to A	3	48^18			1829541	532030568071	946613817344
4	14^23			229	585692886981	495482220544	23	32					4	14^23			229	585692886981	495482220544
5	19^20			37	589973457545	958193355601	20	31					5	19^20			37	589973457545	958193355601
6	3^50				717897987691	852588770249	50	27					6	3^50				717897987691	852588770249
7	33^15				59938945498	865420543457	15	23					7	33^15				59938945498	865420543457
8	2^70				1180591620	717411303424	70	20					8	2^70				1180591620	717411303424
9	121^10				672749994	932560009201	10	18					9	121^10				672749994	932560009201
10	2^67				147573952	589676412928	67	17					10	2^67				147573952	589676412928
11	272^8				29960650	O73923649536	8	15					11	272^8				29960650	O73923649536
12	79^9				119851	595982618319	9	13					12	79^9				119851	595982618319
13	7^17				232	630513987207	17	12					13	7^17				232	630513987207
14	93^7				60	170087060757	7	10					14	93^7				60	170087060757
15	67^6					90458382169	6	9					15	67^6					90458382169
16	5^13					1220703125	13	8					16	5^13					1220703125
17	2^27					134217728	27	7					17	2^27					134217728
18	256^3					16777216	3	6					18	256^3					16777216
19	13^4					28561	4	5					19	13^4					28561
20	2^12					4096	12	4					20	2^12					4096
21	43^2					1849	2	3					21	43^2					1849
22	2^5					32	5	2					22	2^5					32
343368382029251	248465784908	928111829807	2487564842145	7920283863343									343368382029251	248465784908	928111829807	2487564842145	7920283863343	  < Total of individual columns
343368382029251	248465784908	928111829809	487564842152	920283863343	< 63 Digit Palindrome								343368382029251	248465784908	928111829809	487564842152	920283863343	< 63 Digit Palindrome

-	B.S.Rangaswamy
21.09.2009

```

Contributions

B.S. Rangaswamy (email) dd. [ various dates in 2009 ].

```

```