Problems & Puzzles: Puzzles

Puzzle 407. Prime Multidimensional Arithmetic Progressions

Anton Vrba contributes the following puzzle:

Over on mersenneforum.org  the thread http://www.mersenneforum.org/showthread.php?p=105223#post105223  ‘fivemack’ has investigated primes in multidimensional arithmetic progression and has challenged to better his/her results. Reformulating the ‘fivemack’ idea, lets define APn^d, as a arithmetic progression of length n in d dimensions, has n^d elements, and it is unique if no element is duplicated.

 Type Formula APn^1 p = q + a*x x = 0,1,..n-1 APn^2 p = q + a*x +b*y x,y = 0,1,…n-1 APn^3 p = q + a*x +b*y + c*z x,y,z = 0,1,…n-1 APn^4 p = q + a*x +b*y + c*z + d*r x,y,z,r = 0,1,…n-1 APn^5 p = q + a*x +b*y + c*z + d*r + e*s x,y,z,r,s = 0,1,..n-1 APn^5 p = q + a*x +b*y + c*z + d*r + e*s +d*t x,y,z,r,s,t = 0,1,..n-1

APn^1 are the well known and documented primes in arithmetic progression. ‘fivemack’ gives following examples for higher dimensions:

AP3^3 : 123401 + 90x + 31590y + 42126z is prime for x,y,z = 0,1,2

AP2^4 : 11 + 6x + 30x + 20y +42r  is prime for x,y,z,r = 0,1

AP2^5 : 43 + 66x + 120y + 270z + 340r + 378s is prime for x,y,z,r,s = 0,1

and

22123 + 330x + 26550y + 26880z is an example of a non-unique AP3^3

Questions:

1. Find the smallest prime APn^d for  d= 2,3,…, n=2,3,…

2. Find the largest prime  APn^d for  d= 2,3,…, n=2,3,…

Frederick Schneider wrote:

I found the minimum answer for AP4^2. I defined minimum as the
minimax prime in the set of primes found by

2089 is the minimax solution where duplicates are allowed
199 + 210x + 420y for x=0..3 and y= 0..3

5237 is the minimax solution where duplicates are not allowed
503 + 360x + 1218y for x=0..3 and y= 0..3

***

Jan van Delden wrote:

AP2^2: 3+2x+8y  x,y in [0,1] minimax

AP3^2: 5+12x+42y x,y in [0,1,2] minimax?

AP4^2: 7+100062x+209220y x,y in [0,1,2,3] minimax

no duplicates

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