Puzzle 1024. Primes from primes

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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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Adam wrote:

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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