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

1,1

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

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 *)

Crossrefs

Cf. A057015, A125780.

Sequence in context: A190385 A201251 A235097 * A233925 A032752 A104928

Adjacent sequences: A125776 A125777 A125778 * A125780 A125781 A125782

Keyword

nonn

Author

Artur Jasinski, Dec 09 2006

Extensions

Corrected and extended by Donovan Johnson, Apr 22 2008

Status

approved