A224197 Least b > p_n^2 such that [p_1^2,p_2^2,...,p_n^2] in base b is prime, where p_j denotes the j-th prime.

11, 26, 51, 124, 177, 312, 394, 668, 843, 978, 1398, 1730, 1911, 2242, 2859, 3496, 3724, 4532, 5073, 5358, 6269, 6906, 7927, 9422, 10205, 10766, 11522, 12060, 12923, 16142, 17220, 18788, 19409, 22806, 22965, 25562, 26570, 28038, 30636

Offset: 2

Author

Zhi-Wei Sun, Apr 01 2013

Keywords

• nonn

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).

Examples

			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.
		

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.

Crossrefs

Cf. A000040, A217788, A218465, A224210.

Programs

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}]]