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A110974 Numbers n such that 23*n^2 - 1 is prime. 0

%I #15 Sep 08 2022 08:45:20

%S 4,6,8,14,18,20,22,38,52,60,62,64,84,90,92,94,108,126,130,134,140,146,

%T 148,150,168,172,176,178,192,202,220,224,242,286,290,300,304,308,316,

%U 326,328,344,346,350,354,360,378,396,398,400

%N Numbers n such that 23*n^2 - 1 is prime.

%C Let's look at the sequence in base 12 with X for ten and E for eleven. Recall that in base 12 all primes greater than 3 end in either 1, 5, 7, or E. The sequence [n,(23*n^2-1) mod 12], 0<=n<=11, is [0, E], [1, 10], [2, 7], [3, 2], [4, 7], [5, 10], [6, E], [7, 10], [8, 7], [9, 2], [10, 7], [11, 10] so the only possibilities for primes are at the even integers with 7 and E primes, with 7 primes at 2, 4, 8, X mod 12 and E primes at 0, 6 mod 12. The sequence in base 12 is [4,267], [6,58E], [8,X27], [12,2737], [16,438E], [18,53X7], [1X,6537], [32,17277], [44,2EEX7], [50,3EXEE], [52,431E7], [54,46627], [70,79XEE], [76,8E98E], [78, 947X7], [7X,99737], [90,10E2EE]. - _Walter Kehowski_, Oct 05 2005

%e If n=94 then 23*n^2 - 1 = 203227 (prime).

%p select(proc(z) isprime(z[2]) end, [seq([n,23*n^2 - 1],n=0..9*12)]); # _Walter Kehowski_

%t Select[Range[400], PrimeQ[(23#^2 - 1)] &] (* _Stefan Steinerberger_, Feb 28 2006 *)

%o (Magma) [ n: n in [0..1500] | IsPrime( 23*n^2 - 1) ] // _Vincenzo Librandi_, Jan 31 2011

%o (PARI) isok(n) = isprime(23*n^2 - 1); \\ _Michel Marcus_, Sep 16 2015

%K nonn

%O 1,1

%A _Parthasarathy Nambi_, Sep 27 2005

%E More terms from _Stefan Steinerberger_, Feb 28 2006

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Last modified March 5 09:09 EST 2024. Contains 370545 sequences. (Running on oeis4.)