A065428 Numbers k such that no x^2 mod k is prime.

1, 2, 3, 4, 5, 8, 12, 15, 16, 24, 28, 40, 48, 56, 60, 72, 88, 112, 120, 168, 232, 240, 280, 312, 408, 520, 760, 840, 1320, 1848

Offset: 1

Author

Joerg Arndt, Nov 16 2001

Keywords

• nonn
• nice
• hard
• more

Comments

All numbers in this sequence except 56 are idoneal (A000926) - Joerg Arndt, Jul 13 2005

No more terms < 10^6. - T. D. Noe, Aug 10 2007

No more terms < 10^11. - Charles R Greathouse IV, Dec 15 2008

Numbers x such that all x^3 mod k are nonprimes are 1, 2, 7, 9, 63, and apparently no more.

Links

• Joerg Arndt, Matters Computational (The Fxtbook), p. 784

Crossrefs

Cf. A179402 (x^4 mod n).

Cf. A010051, A000196, A000290.

Cf. A214583 (n such that for all k with gcd(n, k) = 1 and n > k^2, n - k^2 is prime).

Programs

Haskell

a065428 n = a065428_list !! (n-1)
a065428_list = filter f [1..] where
   f x = all (== 0) $
         map (a010051' . (`mod` x) . a000290) [a000196 x .. x-1]
-- Reinhard Zumkeller, Aug 01 2012, Aug 15 2011

Mathematica

t={}; Do[s=Union[Mod[Range[n]^2,n]]; If[Select[s,PrimeQ]=={}, AppendTo[t,n]], {n,1000}]; t  (* T. D. Noe, Aug 10 2007 *)
nx2pQ[n_]:=Module[{m=PowerMod[Range[3n],2,n]},Count[ FindTransientRepeat[ m,2][[2]], ?PrimeQ]==0]; Select[Range[2000],nx2pQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jun 11 2019 *)

PARI

for(n=1, 10^9, q=1; for(x=1, n-1, if(isprime(lift(Mod(x,n)^2)), q=0; break())); if(q, print1(n, ", "))); \\ edited, Joerg Arndt, Jan 28 2015

Python

from sympy import isprime
def ok(n): return not any(isprime((x**2)%n) for x in range(2, n))
print(list(filter(ok, range(1, 2000)))) # Michael S. Branicky, May 08 2021