A125779 Numbers n such that n^4 + 1, n^4 + 3, n^4 + 7 and n^4 + 9 are all prime.

83270, 519370, 939220, 1844170, 2263910, 2293460, 2429260, 2595980, 3133640, 3216530, 3474200, 3559760, 4787050, 5306720, 5505940, 6238780, 6889430, 6932770, 7320160, 8286340, 8427880, 8744290, 8961590, 9863440, 10871530

Offset: 1

Author

Artur Jasinski, Dec 09 2006

Keywords

• nonn

Comments

Schinzel proved in 1958 that the set of primes of kind n^(2^k) + 1, n^(2^k) + 3, n^(2^k) + 7 and n^(2^k) + 9 is infinite for each number k > 0.

References

• Sierpinski, W. Elementary theory of numbers. Warszawa 1964 Monografie Matematyczne Vol. 42.

Links

• Robert Israel, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A057015, A125780.

Programs

Maple

R:= NULL: count:= 0:
for k from 0 while count < 30 do
  for i in [20,30,40,50] do
    n:= 70*k + i;
    if isprime(n^4+1) and isprime(n^4+3) and isprime(n^4+7) and isprime(n^4+9) then
      count:= count+1; R:= R, n;
    fi
od od:
R; # Robert Israel, Feb 11 2021

Mathematica

Select[Range[109*10^5],AllTrue[#^4+{1,3,7,9},PrimeQ]&] (* Harvey P. Dale, May 15 2022 *)

Extensions

Corrected and extended by Donovan Johnson, Apr 22 2008