| World!Of Numbers |  |
||
 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) 
Palindromic Sums of Powers
| The powers expression | The palindromic curio | Length |
|---|---|---|
| B.S.Rangaswamy [ June 15, 2009 ] curio 9 | ||
| 605 + 533 + 1702 | 777777777 | 9 |
| B.S.Rangaswamy [ July 11, 2009 ] curio 10 - minimum one 3rd, 4th or higher power | ||
| 230 + 61132 + 172 + 152 + 22 | 1111111111 | 10 |
| 231 + 86452 + 502 + 72 | 2222222222 | 10 |
| 577352 + 502 + 242 + 25 | 3333333333 | 10 |
| 16443 + 10702 + 113 + 152 + 22 | 4444444444 | 10 |
| 129 + 198942 + 153 + 242 + 24 | 5555555555 | 10 |
| 18823 + 913 + 1102 + 33 | 6666666666 | 10 |
| 881912 + 503 + 142 + 102 | 7777777777 | 10 |
| 233 + 172902 + 104 + 142 | 8888888888 | 10 |
| 999992 + 4472 + 53 + 26 | 9999999999 | 10 |
| B.S.Rangaswamy [ July 11, 2009 ] curio 20 | ||
| 33333333332 + 326 + 61132 + 172 + 152 + 22 | 11111111111111111111 | 20 |
| 47140452072 + 926422 + 203 + 54 + 222 + 102 | 22222222222222222222 | 20 |
| 57735026912 + 1017302 + 463 + 402 + 24 | 33333333333333333333 | 20 |
| 66666666662 + 942802 + 553 + 632 + 122 | 44444444444444444444 | 20 |
| 74535599242 + 1220512 + 4722 + 152 + 132 | 55555555555555555555 | 20 |
| 81649658092 + 672872 + 105 + 402 + 63 | 66666666666666666666 | 20 |
| 88191710362 + 1247252 + 513 + 782 + 112 | 77777777777777777777 | 20 |
| 94280904152 + 1243942 + 533 + 202 + 53 + 52 | 88888888888888888888 | 20 |
| 99999999992 + 1414212 + 105 + 262 + 34 OR 46415883 + 73402262 + 11222 + 182 + 35 | 99999999999999999999 | 20 |
Raw data
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
Please find herewith enclosed 'Curio 32' for your kind perusal.
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
> I'll ask you to proofread it and then it can get uploaded.
>
> 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
> >
> > Your sincerely
> >
> > 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.
Your Sincerely
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
Your sincerely
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
Grandsons 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 ].
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Patrick De Geest - Belgium
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