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Palprimes in 'Arithmetic Progression'

Harvey Dubner (†) sent me solutions for **5**, **6**, **8**, **9** and **10** palprimes in *arithmetic progression*.

With this great contribution Harvey consolidates his position as the number one master in palindromic primes.

Palindromic 'Sophie Germain' Primes

More extensive info was available in Warut Roonguthai's (discontinued) site about

Palindromic Sophie Germain Primes

and Yves Gallot's Proth.exe and Cunningham Chains

On [ *April 21, 1999* ] I received the following email from Harvey Dubner.

"I cannot resist looking for palindromic primes that have other interesting properties.

Naturally, I have to share the results with somebody, so here they are.

P, Q, R are palindromic primes. Q = 2P+1, R = 2Q+1.

Thus P, Q are Sophie Germain primes.

P, Q, R is also a Cunningham chain of Palindromic primes.

There cannot be a fourth palindromic member of the chain.

Smallest possible set - from

“J. Recreational Math.”, Vol.26(1), 1994,

“Palindromic Sophie Germain primes” by Harvey Dubner

23 digits

P = 19091918181818181919091 Q = 38183836363636363838183 R = 76367672727272727676367Largest known set, recently found March 22, 1999

Also, number of digits is a palindromic prime.

Also, P is a tetradic prime (4-way prime)

P, Q, R all have 727 digits, the number of digits is a palindromic prime.

P = 1818181808080818080808180818081818081808181818081808080818180808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080808080808080808080808080808080808_ 0808080808080808080808080808080808080818180808081808181818081808181808_ 180818080808180808081818181 Q = 3636363616161636161616361636163636163616363636163616161636361616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161616161616161616161616161616161616_ 1616161616161616161616161616161616161636361616163616363636163616363616_ 361636161616361616163636363 R = 7272727232323272323232723272327272327232727272327232323272723232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323232323232323232323232323232323232_ 3232323232323232323232323232323232323272723232327232727272327232727232_ 723272323232723232327272727Sorry, there is no shorthand way of expressing these palprimes.

Harvey Dubner"

Smallest Palprimes in 'Arithmetic Progression'

by Warut Roonguthai [ *June 21-24, 1999* ]

Here are the 5 smallest arithmetic progressions of **3**, **4**, **5**, **6** and **7**

palindromic primes (ranked by the size of the last term):

Between brackets are the 'common difference' values.

**3**

**3**, **5**, **7** (2)

**3**, **7**, **11** (4)

**11**, **101**, **191** (90)

**727**, **757**, **787** (30)

**10501**, **12421**, **14341 **(1920)

**4**

**13931**, **14741**, **15551**, **16361** (810)

**10301**, **13331**, **16361**, **19391** (3030)

**73637**, **75557**, **77477**, **79397** (1920) - not mentioned by Beiler

**70607**, **73637**, **76667**, **79697** (3030)

**94049**, **94349**, **94649**, **94949** (300)

**5**

**1150511**, **1262621**, **1374731**, **1486841**, **1598951** (112110)

**1114111**, **1335331**, **1556551**, **1777771**, **1998991** (221220)

**7190917**, **7291927**, **7392937**, **7493947**, **7594957** (101010)

**9185819**, **9384839**, **9583859**, **9782879**, **9981899** (199020)

**9585859**, **9686869**, **9787879**, **9888889**, **9989899** (101010)

**6**

**10696969601**,
**11686968611**,
**12676967621**,
**13666966631**,
**14656965641**,
**15646964651**
(989999010)

**12374047321**,
**13364046331**,
**14354045341**,
**15344044351**,
**16334043361**,
**17324042371**
(989999010)

**13308880331**,
**14417771441**,
**15526662551**,
**16635553661**,
**17744444771**,
**18853335881**
(1108891110)

**19125452191**,
**19135553191**,
**19145654191**,
**19155755191**,
**19165856191**,
**19175957191**
(10101000)

**14282128241**,
**15272127251**,
**16262126261**,
**17252125271**,
**18242124281**,
**19232123291**
(989999010)

**7** - Three example of 7 palindromic primes in AP.

Warut said it's very likely that the first two are the smallest ones.

Some Straightforward 'Plateau' Palprimes

by Nicholas Angelou [ *July 8, 2000* ]

Below are some prime numbers that are the form of **1kkk...k1**

starting with 1 and ending with 1 and ALL the in_between digits

are the SAME digit **2** to **9** in repetition)

* N-1 ,Brillhart-Lehmer-Selfridge proving algorithm18...81 9 digits

19...91 9 digits

16...61 13 digits

18...81 15 digits

16...61 17 digits

16...61 19 digits

15...51 21 digits *

15...51 33 digits *

16...61 37 digits *

17...71 49 digits *

16...61 73 digits *

For the full listing of these kind of primes (PDP's) see Plateau and Depression Primes

Palindromic Prime Pyramid Puzzle

by G. L. Honaker, Jr. [ *January 20, 2000* ]

2 727 37273 333727333 93337273339 309333727333903 1830933372733390381 92183093337273339038129 3921830933372733390381293 1333921830933372733390381293331 18133392183093337273339038129333181 *** *** *** |
---|

The puzzle is quite simple in fact.

Can you extend the pattern beyond these **11** rows up to let's say **727** (cf. row 2) or even **929** ?

Note that each term is the smallest to have the previous term as a centered substring,

beginning with the smallest palindromic prime 2.

Garland and I believe that working on this problem is the best cure for 'aibohphobia'

or 'fear of palindromes'...

Here is Carlos Rivera's comment from [ *January 22, 2000* ]

I have started getting the solution to this sequence. Inside the capabilities

of UBASIC this sequence can be obtained only up to 400 rows or so... why ?

I'm now at row 100 and the number is 563 digits large (of course that I'm only

printing the additional substring from one number to the other, and also the

quantity of digits of each number).

Then the target of 929 rows is out of possibilities to any known public code

available: this number (929 rows) must be no less than 4500 digits large,

if it exists...

Needless to say that all the numbers in my sequence are, by the moment

"strongpseudoprime". The bad news is that any mistake in any step spoils the

rest of the numbers... so, this is a very hard puzzle in time consumption

terms, before we can be sure to say that all the members are palprimes.

Watch it!...

And here is Carlos Rivera's message from the day after [ *January 23, 2000* ]

If you're interested, the complete sequence can be consulted via the "

Good sunday. I have stopped my search at 1001 digits.

Now the search has begun to be slow. I let other (brave) people continue...row extreme total number left digits added 1 2 1 2 7 3 3 3 5 4 33 9 5 9 11 6 30 15 ... 158 1156 957 159 1011 965 160 38 969 161 1156 977 162 1070 985 163 1419 993 164 1459 1001

menu-function of most modern browsers applied to this very page (see then list at the end of that page).

Chris Caldwell and G. L. Honaker, Jr. wrote an article that soon will be publicized

into the

to come by as a preview of their discussion about these types of pyramids can be found

at the following web-addresses :

3131 11311 121131121 1212113112121 36121211311212163 303612121131121216303 7230361212113112121630327 30723036121211311212163032703 723072303612121131121216303270327 1472307230361212113112121630327032741 114723072303612121131121216303270327411 7711472307230361212113112121630327032741177 1237711472307230361212113112121630327032741177321 *** *** *** |

About palprime '13331'

It shouldn't be too hard to find a few more like palprimes ! Please submit them to me.103030301= 11 x 29 x 71 x 4549

113131311= 3 x 17 x 37 x 167 x 359

123232321= 23 x 2239 x 2393

133333331= 11287 x 11813

143434341= 3 x 3 x 3 x 293 x 18131

153535351= 1997 x 76883

163636361= 7 x 23376623

173737371= 3 x 4517 x 12821

183838381= 13469 x 13649

193939391= 79 x 2454929

In case this assignment is too easy for you...

what about finding the first palprime with this counter-property :

*Inserting any digit d between adjacent digits of palprime P always produces a new prime !*

Take this opportunity to become eternally praised and find this elusive palprime P !

Your name will always be linked with this extraordinary number, if it exists of course !

My best shot up to now [ *October 3, 1999* ] is this '**6 out of 10**' solution with palprime **131**,

but was quickly superseeded by Carlos Rivera's '**7 out of 10**' 13-digit solution.

Thanks to Carlos Rivera we now have this '10301= prime

11311= prime

12321= 3 x 3 x 37 x 37

13331= prime

14341= prime

15351= 3 x 7 x 17 x 43

16361= prime

17371= 29 x 599

18381= 3 x 11 x 557

19391= prime

On [7762868682677_0= prime

7762868682677_1= 29 x 401 x 10038749783 x 61432020031

7762868682677_2= prime

7762868682677_3= 11 x 9411043 x 123490751 x 576788519

7762868682677_4= prime

7762868682677_5= prime

7762868682677_6= prime

7762868682677_7= prime

7762868682677_8= prime

7762868682677_9= 10987 x 33599 x 21616267514286769

of 19 digits :

This still means we need an '

Enjoy the search !0 out of 10=13331

1 out of 10=101

2 out of 10=383

3 out of 10=151

4 out of 10=11311

5 out of 10=11(was first353but Jim Howell corrected this)

6 out of 10=131

7 out of 10=7762868682677by Carlos Rivera [October 20, 1999]

8 out of 10= ?

9 out of 10= ?

10 out of 10= ?

Solutions may also be forwarded to my friend Carlos Rivera (email)

as he maintains a similar puzzle page about this topic.

Please visit his PP&P puzzle 72 on Persistent Palprimes.

Featured in Prime Curios! 13331

About palprime '134757431'

This paragraph deals with another yet very unique palindromic prime.

The number in question is **134757431**

It can be expressed as a ninedigits sequence raised to the power of another combination of the ninedigits sequence.

And that in exactly three different ways. No other palindrome (even a composite one) exists with that triple property !

134757431 1 ^{7}+ 2^{3}+ 3^{8}+ 4^{5}+ 5^{4}+ 6^{2}+ 7^{1}+ 8^{9}+ 9^{6}1 ^{7}+ 2^{5}+ 3^{8}+ 4^{1}+ 5^{2}+ 6^{4}+ 7^{3}+ 8^{9}+ 9^{6}1 ^{7}+ 2^{8}+ 3^{4}+ 4^{2}+ 5^{3}+ 6^{5}+ 7^{1}+ 8^{9}+ 9^{6}

Another unique palindrome prime exists that is expressible like explained above but only in a twofold manner.

You like math puzzles ? Ok, then find that palindromic prime !

Some smallest primes that turn out to be palindromic !

Featured in Prime Curios! 1666666666661

**Carlos Rivera Jaime Ayala**

Magic Squares composed of only palprimes

A collaboration of Carlos Rivera and Jaime Ayala turned out to be very fruitful as after

two or three weeks of exhaustive searching they came up with the following two extraordinary

Magic Squares [ *May 22** and May 24,1999* ]. __Quoting__

"As far as our search is correct these two are the smallest Magic Squares with palprimes."

We found no solutions with 3, 5, 7 or 9 digits and no other with 11 digits.

Due to the work of Jean Claude Rosa [ *March 31, 2002* ]

we know that Rivera's and Ayala's Magic Squares are not the smallest ones.

For all the details see WONplate 129.

10915551901 | 12133533121 | 11527872511 | Magic Square I |
---|---|---|---|

12137973121 | 11525652511 | 10913331901 | all numbers are palprimes |

11523432511 | 10917771901 | 12135753121 | by Rivera & Ayala |

10797779701 | 14336063341 | 12568586521 | Magic Square II |

14338283341 | 12567476521 | 10796669701 | all numbers are palprimes |

12566366521 | 10798889701 | 14337173341 | by Rivera & Ayala |

Palprime sums of three consecutive palprimes

Carlos Rivera from Nuevo León, México constructed the following nice series [ *May 26, 1998* ],

displaying sums of three consecutive palindromic primes that are also palindromic primes.

Many more no doubt can be found. Select the most beautiful ones and send them to me, please !

See also [OEIS A046492.]

101 + 131 + 151 = 383 30103 + 30203 + 30403 = 90709 31013 + 31513 + 32323 = 94849 1221221 + 1235321 + 1242421 = 3698963 102838201 + 103000301 + 103060301 = 308898803 110111011 + 110232011 + 110252011 = 330595033 111010111 + 111020111 + 111050111 = 333080333 133020331 + 133060331 + 133111331 = 399191993 302313203 + 302333203 + 302343203 = 906989609 323222323 + 323232323 + 323333323 = 969787969 (*) 11021312011 + 11021412011 + 11022122011 = 33064846033 12301010321 + 12303230321 + 12303730321 = 36907970963 13002220031 + 13002420031 + 13004040031 = 39008680093 13100100131 + 13100300131 + 13101410131 = 39301810393 and 8 more from Chai Wah Wu [Sep 12, 2019] 30000500003 + 30002320003 + 30005150003 = 90007970009 32222122223 + 32223132223 + 32224342223 = 96669596669 ? + ? + ? = 3003899983003 ? + ? + ? = 3006918196003 ? + ? + ? = 3033959593303 ? + ? + ? = 3090048400903 ? + ? + ? = 3306986896033 ? + ? + ? = 3336904096333 Can you continue to replace the questionmarks ?

Note that the middle number of the sum **(*)** is a special number !

It is in fact an undulating palindromic prime.

Click the href link to find out more info !

The next trio of consecutive palindromic primes - 43 digits - is impressive...

1000000000000002109952599012000000000000001 +
1000000000000002110000000112000000000000001 + 1000000000000002110025200112000000000000001 = 3000000000000006329977799236000000000000003 |

The above and following examples are the smallest possible ones for that given length because they start form a "zero nut"

(the "nut" is the central number before "reflecting" it to get the central palindrome condition and at the same time the overall palindrome condition... do you follow me?)

and the nut is then increased one by one until the three palprimes added yield another palprime.

... but still nothing compared to these awesome - 99 and 191 digits - constructions (both by Carlos Rivera) !

Note, it took Carlos one week to get the **191** (itself also a palprime!) solution with his code !

1(0)
_{42}3332222222333(0)_{42}1 +1(0) _{42}3332253522333(0)_{42}1 +1(0) _{42}3332304032333(0)_{42}1 =3(0) _{42}9996779776999(0)_{42}3 |

1(0)
_{87}132298010892231(0)_{87}1 +1(0) _{87}132300858003231(0)_{87}1 +1(0) _{87}132301111103231(0)_{87}1 =3(0) _{87}396899979998693(0)_{87}3 |

See also Carlos Rivera's ** Puzzle nr. 23** : Pal-primes adding consecutive primes.

[ *June 10, 2003* ]

Jens Kruse Andersen (email) found a much larger solution (515 digits).

Its palindromic length is just a bonus!

For the strategy of his clever approach I refer to
Carlos Rivera's webpage at http://www.primepuzzles.net/puzzles/puzz_023.htm.

1(0)
_{243}101031223000000000322130101(0)_{243}1 +1(0) _{243}101031223000313000322130101(0)_{243}1 +1(0) _{243}101031223008545800322130101(0)_{243}1 =3(0) _{243}303093669008858800966390303(0)_{243}3 |

Splendid solution, Jens. Well done!

Exist there a solution with carries occurrence ?

About palprime '71317'

Carlos Rivera discovered that palindromic prime **71317**

is expressible as the sum of consecutive primes in three ways [ *August 9, 1998* ].

71317 2351 + 2357 + .....+ 2579 + 2591 ( 29 primes )10163 + 10169 + 10177 + 10181 + 10193 + 10211 + 10223 ( 7 primes ) 14243 + 14249 + 14251 + 14281 + 14293 ( 5 primes )

Notice that the primes involved in each equation are different,

and that the quantity of primes in each equation is also a prime number.

Featured in Prime Curios! 71317

**Enoch Haga**

Palprimes in the 37^{th} Mersenne prime

is fascinated by the largest known prime (2

**G. L. Honaker, Jr.**

Beasty palprimes in the 37^{th} Mersenne prime

from its power of ten axis are the following two so far

10^{8714} – 1666110 ^{12379} – 1666110 ^{20437} – 16661 |

Trinity of the Beast

One day I wrote an email to a group of people and presented them the sum of the first **666** palindromic primes

namely **2391951273** and asked them if they could relate this to the Number of the Beast.

Very soon afterwards G. L. Honaker, Jr. came up with what he calls the '*Trinity of the Beast*' :

The sum of the first 666 palindromic primes = 2391951273 [De Geest]2 ^{3} + 3^{3} + 9^{3} + 1^{3} + 9^{3} + 5^{3} + 1^{3} + 2^{3} + 7^{3} + 3^{3}= 666 + 666 + 666 [Honaker] |
---|

Note that 3 x **666** = __1998__ is the year in which G. L. Honaker, Jr. made this inspired discovery !

And he continues :

9230329is the666^{th}palindromic prime number.

Featured in Prime Curios! 9230329

9+2+3+0+3+2+9 = 28 (the second perfect number... following 6)

28 = 1^{3}+ 3^{3}... more Trinity of the Beast ?

Note the3middle digits,303, is the number of primes less than or equal to 1999

1999 being the first prime of the form 666n+1 !

Honaker's Constant = __1.323982__0264...

G. L. Honaker, Jr. kindly lends his name to the following 'convergent' infinite series

that is the sum of the reciprocals of all the palindromic primes.

Featured in Prime Curios! 1.32398...

1 2 | + | 1 3 | + | 1 5 | + | 1 7 | + | 1 11 | + | 1 101 | + | 1 131 | + ... = ? |
---|

What is the value of this series ? Garland's very first guess was the square root of two.

This soon turned out not to be the case. Mike Keith came to the rescue as he computed

the value of the sum up through all 11-digit palindromic primes. The value he got was :

1.3239820264... |

Mike commented further

Finally Garland found a nice approximation using a fraction with palprime numerator and denominator :

See more at [OEIS sequences A046852 and A046853.]

Some beautiful palprimes

( This section is best read from bottom to top ! )

A palprime of length 81 ! [

Note

**31415926535897932384626433833462648323979853562951413**

Another beautiful palprime by G. L. Honaker, Jr. [ *March 13, 1999* ]

Using his own terminology we call this a Pi-lindromic prime !

This -lindromic prime has 53 digits. [OEIS references A039954 and A119351.]

Featured in Prime Curios! 31415...51413 (53-digits)

**151515151515151**

A smoothly **undulating** palindromic prime (SUPP) containing 15 digits. [ *March 10, 1999* ]

Hmmm... the smallest 'mnemonic' prime ?

Featured in Prime Curios! 151515151515151

**10080010080010080010080010080010080_
01008001008001008001008001008001008_
00100800100800100800100800100800100_
80010080010080010080010080010080010_
08001008001008001008001008001008001**

A 175-digit tetradic palprime discovered by G. L. Honaker, Jr. on [

Featured in Prime Curios! 10080...08001 (175-digits)

**313_0 _{{150}}_1183811_0_{{150}}_313**

The smallest(?) palprime of length

Note that both

Clearly G. L. Honaker, Jr. is fascinated with triadic (or 3-way) primes i.e. which are invariant upon

reflection only along the line they are written on, so the digits may be 0, 1, 3 and 8.

Featured in Prime Curios! 31300...00313 (313-digits)

**7_0 _{{48}}_666_0_{{48}}_7**

A palprime with a palprime total of digits namely

**31300000000000000000000000006660000000000000000000000000313**

A nice palprime of length 59 sent by G. L. Honaker, Jr. on [ *February 28, 1999* ]

See the section 'Enoch Haga' for the story behind the mysterious number **313** at page 4 .

And yes we all know that **666** is the famous number of the beast...

**Mike Keith**

About 'Descriptive Primes'

Start with a prime (that happens to be palindromic in this case) e.g.

Featured in Prime Curios! 373

Each next term describes the previous terms according to the rules stipulated in A005150.

The puzzle statement is now to find the largest

Our example is the only 3-digit number (a palprime at that!) producing 3 more primes in a row !

He reported to me [

See OEIS sequences [A037033 - A038131 - A038132]

Featured in Prime Curios! 233

Mike Keith has found three more length-6 prime chains [ *February, 1999* ]

Here are the beauties !

Mike Keith predicts a length-7 chain should occur around 10^11, after having made some calculations

about the relative probability to find chains of this type of different length.

[ *November 2002* ]

This is not the end of the story ! An exhaustive search was done by Walter Schneider.

Currently the search has arrived at starting number 10^11. In total 59 sequences of

length 6 are found and one sequence of length 7 starting at 19972667609.

So far no sequence of length 8 or more is known.

Carlos Rivera (email) from Nuevo León, México and Jaime Ayala (email) - go to topic

Enoch Haga (email) for his discoveries of palindromes in the 37^{th} Mersenne prime - go to topic

G. L. Honaker, Jr. (email) from Bristol, Virginia made a beastly interesting discovery - go to topic

Mike Keith (email) found many of the larger self-descriptive primes- go to topic

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Patrick De Geest - Belgium - Short Bio - Some Pictures

E-mail address : pdg@worldofnumbers.com