Problems & Puzzles: Puzzles

 Puzzle 1024. Primes from primes Claudio Meller made me notice this nice curio: 2^3+5^7+...+31^37 = 15148954872646850196557152427604893685308877022260348791 is prime. Q. Can you find more primes of these type.

During the week 19-24. December 2020, we received the following contributions: Paul Cleary, Adam Stinchcombe, Oscar Volpatti, Jan van Delden, Giorgos Kalogeropoulos, Metin Sariyar, Simon Cavegn, Jeff Heleen

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Paul wrote:

If we allow the start Prime to be other than 2, then there are the following solutions of this type.

{{257,263},{269,271},{277,281},{283,293},{307,311},{313,317},{331,337},{347,349},{353,359},{367,373},

{379,383},{389,397},{401,409},{419,421},{431,433}} =

535374867...81339

{{653,659},{661,673},{677,683},{691,701},{709,719},{727,733},{739,743},{751,757},{761,769},{773,787},

{797,809},{811,821},{823,827},{829,839},{853,857},{859,863},{877,881},{883,887},{907,911},{919,929},

{937,941},{947,953},{967,971},{977,983},{991,997},{1009,1013},{1019,1021},{1031,1033},{1039,1049}} =

268973...971857

{{43,47},{53,59},{61,67},{71,73},{79,83},{89,97},{101,103},{107,109},{113,127},{131,137},{139,149},
{151,157},{163,167},{173,179},{181,191},{193,197},{199,211},{223,227},{229,233},{239,241},{251,257},

{263,269},{271,277},{281,283},{293,307},{311,313},{317,331},{337,347},{349,353},{359,367},{373,379},

{383,389},{397,401},{409,419},{421,431},{433,439},{443,449},{457,461},{463,467},{479,487},{491,499},

{503,509},{521,523},{541,547},{557,563},{569,571},{577,587},{593,599},{601,607},{613,617},{619,631},

{641,643},{647,653},{659,661},{673,677},{683,691},{701,709},{719,727},{733,739},{743,751},{757,761},

{769,773},{787,797},{809,811},{821,823},{827,829},{839,853},{857,859},{863,877},{881,883},{887,907},

{911,919},{929,937},{941,947},{953,967},{971,977},{983,991},{997,1009},{1013,1019},{1021,1031},

{1033,1039},{1049,1051},{1061,1063},{1069,1087},{1091,1093},{1097,1103},{1109,1117}} =

154360...677681

{{41,43},{47,53},{59,61},...{1607,1609},{1613,1619},{1621,1627}} =

207474...769879

...

Have taken the primes up to the 4740 th prime with no solution found.

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It wasn't clear to me if the puzzle had to start at 2 or not.  I found, starting at 257^263, including 15 summands, and ending with 431^433, results in the 1141 digit prime:

53537486...0781339

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Oscar wrote:

I initially considered only sums of the "base" form:
x(n) = 2^3 + 5^7 + ... + p(2*n-1)^p(2*n).
I checked all candidates with index n <= 1500, finding no further primes after the known value:
x(6) = 2^3 + 5^7 + ... + 31^37,   56 digits.
The sequence grows very quickly; the largest tested value was:
x(1500) = 2^3 + 5^7 + ... + 27437^27449,   121828 digits.

Then I tried to start from an odd prime p(k) > 2, considering sums of the "extended" form:
y(k,n) = p(k)^p(k+1) + p(k+2)^p(k+3) + ... + p(k+2*n-2)^p(k+2*n-1).
As before, n is the number of addends within the given sum.
I checked all candidates with less than 20000 digits, finding ten PRPs.
I verified primality only for the smallest one.

y(55,15) = 257^263 + 269^271 + ... + 431^433,   1141 digits (prime!);
y(119,29) = 653^659 + 661^673 + ... + 1039^1049,   3165 digits;
y(14,87) = 43^47 + 53^59 + ... + 1109^1117,   3402 digits;
y(13,123) = 41^43 + 47^53 + ... + 1621^1627,   5223 digits;
y(345,85) = 2333^2339 + 2341^2347 + ... + 3673^3677,   13109 digits;
y(134,213) = 757^761 + 769^773 + ... + 4049^4051,   14614 digits;
y(361,129) = 2437^2441 + 2447^2459 + ... + 4549^4561,   16684 digits;
y(307,159) = 2027^2029 + 2039^2053 + ... + 4603^4621,   16927 digits;
y(19,321) = 67^71 + 73^79 + ... + 4933^4937,   18233 digits;
y(596,51) = 4363^4373 + 4391^4397 + ... + 5233^5237,   19476 digits.

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Jan wrote:

A search for all primes where I limited the sum to 25000 digits revealed the following solutions:

 Index Prime From To From To Digits 1 13 2 37 (*) 56 13 258 41 1627 5223 14 187 43 1117 3402 19 660 67 4937 18233 55 84 257 433 1141 119 176 653 1049 3165 134 559 757 4051 14641 307 624 2027 4621 16927 345 514 2333 3677 13109 361 618 2437 4561 16684 596 697 4363 5237 19476 739 752 5623 5711 21451

I also searched for solutions where I reversed the order of the primes in base^exponent.

The smallest solution was the prime:

83^79+97^89+103^101+109^107+127^113+137^131+149^139+157^151+167^163

A table comprising all solutions where the sum has less than 10000 digits:

 Index Prime From To From To Digits 22 39 79 167 (*) 363 27 56 103 263 (*) 622 67 232 331 1459 4598 111 204 607 1249 3831 152 329 881 2207 7367 171 392 1019 2693 9224 185 222 1103 1399 4345 247 416 1567 2861 9876

All numbers are strong pseudo primes, I used 11 Miller-Rabin tests. The primes with an (*) are certified prime.

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Giorgios wrote:

Starting with 2^3..., no other primes were found. (I checked up to 4000 consecutive primes)
If we start from an other prime than 2, here are some results (these sums contain only consecutive primes):

257^263 + ... + 431^433           is a 1141-digit prime (30 consecutive primes)

653^659 + ... + 1039^1049       is a 3165-digit prime (58 consecutive primes)

2333^2339 + ... + 3673^3677   is a 13109-digit prime (170 consecutive primes)

43^47 + ... + 1109^1117            is a 3402-digit prime (174 consecutive primes)

41^43 + ... + 1621^1627           is a 5223-digit prime (246 consecutive primes)

757^761 + ... + 4049^4051       is a 14614-digit prime (426 consecutive primes)

67^71 + ... + 4933^4937           is a 18233-digit prime (642 consecutive primes)

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Metin wrote:

Q1: I didn't find a new prime or prp up to  ...+15413^15427

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Simon wrote:

Found no more solutions up to last term 37747^37781.

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Jeff wrote:

I have checked up to p=499 and q=503 with no further primes.

I have also checked 3 other versions of this puzzle and found only:
2^3+3^5 = 251 prime
3^2+5^3+7^5+11^7+13^11+17^13+19^17+23^19+29^23 = 4316720792370367095095683949638501 prime
3^2+7^5+13^11+19^17+29^23 = 4316720717754896157600395335336781 prime

and no others up to p=499 and q=503.
Jeff Heleen

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