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Prime numbers p such that the Lucas number L(p) can be written in the form a^2 + 2*b^2.
1

%I #32 Jan 04 2022 21:57:33

%S 2,3,5,13,17,29,37,41,53,61,89,97,113,137,157,197,281,313,349,353,397,

%T 433,457,461,509,541,557,593,613,617,661,673,809,829,857,877,1061,1097

%N Prime numbers p such that the Lucas number L(p) can be written in the form a^2 + 2*b^2.

%C These Lucas numbers L(p) have no prime factor congruent to 5 or 7 (mod 8) to an odd power.

%H Blair Kelly, <a href="http://mersennus.net/fibonacci">Fibonacci and Lucas factorizations</a>

%o (PARI) forprime(i=2, 500, a=factorint(fibonacci(i-1)+fibonacci(i+1))~; has=0; for(j=1, #a, if(a[1, j]%8>4&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==1, print(i", ")))

%Y Cf. A000032, A215812.

%K nonn

%O 1,1

%A _V. Raman_, Aug 23 2012

%E 18 more terms from _V. Raman_, Aug 28 2012