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Numbers n such that 23*n^2 + 4 is prime.
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%I #19 Dec 31 2024 15:40:05

%S 3,9,15,57,63,69,75,81,87,99,117,129,141,147,153,177,207,219,273,285,

%T 291,309,327,351,363,375,405,411,453,465,477,483,489,537,543,561,615,

%U 621,627,639,663,699,753,759,789,795,801,831,837,867,873,915,933,939

%N Numbers n such that 23*n^2 + 4 is prime.

%C Are all the numbers in this sequence multiples of 3?

%C Look at the sequence in base 12, with X for ten and E for eleven. Recall that all primes greater than 3 end in 1, 5, 7, E. The sequence [n,(23*n^2+4) mod 12], 0<=n<=11, is [0, 4], [1, 3], [2, 0], [3, 7], [4, 0], [5, 3], [6, 4], [7, 3], [8, 0], [9, 7], [10, 0], [11, 3] so the only possible primes are at 3, 9 mod 12 or only at odd multiples of 3, with the primes being all 7 primes. In base 12 the sequence is [3,157], [9,10E7], [13,2EE7], [49,372E7], [53,449E7], [59,53457], [63,62X57], [69,733E7], [73,848E7], [83,XX557]. - _Walter Kehowski_, Oct 05 2005

%e If n=99 then (23*n^2) + 4 = 225427 (prime).

%p select(proc(z) isprime(z[2]) end, [seq([n,23*n^2 + 4],n=0..9*12)]); (Kehowski)

%o (Magma) [n: n in [0..10000] |IsPrime((23*n^2)+4)]; // _Vincenzo Librandi_, Nov 13 2010

%o (PARI) is(n)=isprime(23*n^2+4) \\ _Charles R Greathouse IV_, Jun 13 2017

%K nonn,easy

%O 1,1

%A _Parthasarathy Nambi_, Sep 26 2005