Problems & Puzzles: Puzzles

Problem 84. Follow-up to problem 67

On April 8, 2023, Jaroslaw Wroblewski sent the "first known prime solution to the Prouhet–Tarry–Escott problem of degree 11"

[32058169621, 32367046651, 32732083141, 33883352071,
34585345321, 35680454791, 36915962911, 38011072381,
38713065631, 39864334561, 40229371051, 40538248081]
=
[32142408811, 32198568271, 32900561521, 33658714231,
34978461541, 35315418301, 37280999401, 37617956161,
38937703471, 39695856181, 40397849431, 40454008891]

All the 24 above terms are (11-digit) prime numbers and the sum of
k-th powers of left terms is equal to the sum of k-th powers of right
terms for any k=1,2,3,...,11.

This solution corresponds to one of the two questions from the Problem 67, from this site, probably posed during the year 2016 or shortly after.

Let's remember the status of the problem 67 in these days and the two mentioned questions:

"In the following table I have extracted some of these solutions (I selected the minimal primes solutions from the results compiled by Shuwen)

 k Author [ai] = [bi], i=1 to n, n=k+1 1 Unknown [ 3, 7 ] = [ 5, 5 ] 2 A. H. Beiler, <1964 [ 43, 61, 67 ] = [ 47, 53, 71 ] 3 Carlos Rivera, 1999 [ 59, 137, 163, 241 ] = [ 61, 127, 173, 239 ] 4 Chen Shuwen, 2016 [ 401, 521, 641, 881, 911 ] = [ 431, 461, 701, 821, 941 ] 5 Qiu Min, Wu Qiang, 2016 [ 17, 37, 43, 83, 89, 109 ] = [ 19, 29, 53, 73, 97, 107 ] 6 Qiu Min, Wu Qiang, 2016 [ 83, 191, 197, 383, 419, 557, 569 ] = [ 89, 149, 263, 317, 491, 503, 587 ] 7 Chen Shuwen, 2016 [ 10289, 14699, 27509, 41579, 42839, 65309, 68669, 77699 ] = [ 10709, 13859, 29399, 36749, 46829, 63419, 70139, 77489 ] 8 Chen Shuwen, 2016 [ 3522263, 4441103, 5006543, 7904423, 9388703, 11897843, 13876883, 15361163, 15643883 ] = [ 3698963, 3981683, 5465963, 7445003, 9954143, 11438423, 14336303, 14901743, 15820583 ] 9 Chen Shuwen, 2016 [ 2589701, 2972741, 6579701, 9388661, 9420581, 15740741, 15772661, 18581621, 22188581, 22571621 ] = [ 2749301, 2781221, 6835061, 8399141, 10314341, 14846981, 16762181, 18326261, 22380101, 22412021 ] 10 ??? Not known 11 ??? Not known

I asked Chen Shuwen to suggest a question about the ideal prime solutions. This was his suggestion:

Q. Send your prime solution for j=1 to k, for k=10 and k=11."

"The above solution is obtained from
[-151, -140, -127, -86, -61, -22, 22, 61, 86, 127, 140, 151]
=
[-148, -146, -121, -94, -47, -35, 35, 47, 94, 121, 146, 148]
by applying linear transformation f(n)=q*n+r,
where
q = 28079730
r = 36298208851

The above solution is related to the following websites:

1) Ideal prime solutions by Chen Shuwen:
http://eslpower.org/eslp.htm#IdeaPrimeSolutions

2) The Prime Puzzles and Problem Connection by Carlos Rivera:
https://www.primepuzzles.net/problems/prob_067.htm

3) Computing Minimal Equal Sums Of Prime Like Powers by Jean-Charles Meyrignac:
Prime Solution Table - prime solution (11,12,12) improves current
record of (11,12,13):
http://euler.free.fr/prime.htm

So far I have found total of 10 solutions to the above problem, the
other 9 being obtained by the following (q,r) pairs:
(28304570,104814825237)
(290176460,142093844439)
(100019290,266218735353)
(467746440,840679980829)
(381293430,1292788605431)
(331759260,1380907662661)
(14407680,2407238347117)
(485107560,2605171968037)
(74070080,6915823849227)

Moreover for the following 2 pairs (q,r):
(83917190,11852078211)
(379276030,36419500473)
we get solutions with some negative terms."

I also asked Jarek if the case k=10 was still unsolved. His answer was: "As far as I know the case k=10 is unsolved in integers, forget the primality.", so...

Q. Solve the case for k=10, no matter the kind of integers you get; but primes are better, of course!

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