OFFSET
2,1
COMMENTS
Conjecture: (i) For any positive integer k and distinct positive integers a_1< a_2 < ... < a_n with a_n prime, there are infinitely many integers b > a_n^k such that [a_1^k,a_2^k,...,a_n^k] in base b is prime.
(ii) For positive integers k, m and n>m, let s_k(m,n) denote the smallest integer b > p_n^k such that [p_m^k,p_{m+1}^k,...,p_n^k] in base b is prime. Then we have the inequality s_k(m,n) <= (n+1)^k*(m+n+1)^k.
This is the k-th power version of the author's conjecture related to A217788. Note that s(m,n) defined there is identical with s_1(m,n). It seems that s_2(m,n) < p_{n+1}*p_{m+n+1}.
For example, [2^2,6^2,9^2,20^2,29^2] in base 900 and [37^2,38^2,60^2,90^2,101^2] in base 10268 are both prime. Also, s_3(1,15) = 103960 and s_5(3,5) = 161098.
Note that for any integer b>13^2 the number [2^2,5,6,156,13^2] in base b is composite since
4x^4+5x^3+6x^2+156x+169 = (4x+13)*(x^3-2x^2+8x+13).
Although 1, 2, 3, 113, 115 are pairwise relatively prime, [1,2,3,113,115] in any base b>115 is composite since x^4+2x^3+3x^2+113x+115 = (x+5)*(x^3-3x^2+18x+23).
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 2..300
Zhi-Wei Sun, A general conjecture involving k-th powers, a message to Number Theory List, April 1, 2013.
EXAMPLE
a(35) = s_2(1,35) = 22806 since [p_1^2,p_2^2,...,p_{35}^2] in base 22806 is prime. Note that p_{36}^2 = 22801 < 22806 < p_{35}*p_{37} = 23393 < p_{36}*p_{37} = 23707.
a(287) = s_2(1,287) = 3519434 since [p_1^2,p_2^2,...,p_{287}^2] in base 3519434 is prime. Note that p_{287}*p_{289} = 3519367 < 3519434 < p_{288}^2 = 3523129 < p_{288}*p_{289} = 3526883.
MATHEMATICA
A[n_, x_]:=A[n, x]=Sum[Prime[k]^2*x^(n-k), {k, 1, n}]
Do[Do[Do[If[PrimeQ[A[n, b]]==True, Print[n, " ", b]; Goto[aa]], {b, Prime[n]^2+1, Prime[n+1]Prime[n+2]-1}];
Print[n, " ", counterexample]; Label[aa]; Continue, {n, 2, 100}]]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 01 2013
STATUS
approved