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A282997
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Primes of the form (p^2 + q^2)/2 such that |q^2 - p^2| is square, where p and q are prime.
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1
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17, 97, 16561, 89041, 2579199841, 3497992081, 5645806321, 21103207681, 428888025121, 686770904161, 2726023770721, 4017427557361, 6831989588161, 6933052766641, 10138513506001, 19387278797041, 23452359542401, 35287577206801, 40057354132561, 62093498771041, 64116963608881
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OFFSET
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1,1
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COMMENTS
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Primes of the form x^4 + y^4 such that q = x^2 + y^2 and p = |y^2 - x^2| are both primes.
Primes of the form n^4 + (n+1)^4 such that q = n^2 + (n+1)^2 and p = 2n+1 are both primes; so for n in A128780.
Primes of the form x^4 + y^4 such that |y^4 - x^4| is a semiprime.
{q, p, a(n) = (p^2+q^2)/2}
{5, 3, 17}
{13, 5, 97}
{181, 19, 16561}
{421, 29, 89041}
{71821, 379, 2579199841}
{83641, 409, 3497992081}
{106261, 461, 5645806321}
{205441, 641, 21103207681}
{926161, 1361, 428888025121}
{1171981, 1531, 686770904161}
(End)
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LINKS
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FORMULA
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a(n) == 1 (mod 16).
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EXAMPLE
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17 = (3^2 + 5^2)/2 and 5^2 - 3^2 = 4^2.
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MATHEMATICA
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lst = {}; a = 2; While[a < 2501, b = Mod[a, 2] + 1; While[b < a, If[ PrimeQ[a^4 + b^4] && PrimeOmega[a^4 - b^4] == 2, AppendTo[lst, (a^4 + b^4)]]; b += 2]; a++]; lst (* Robert G. Wilson v, Feb 27 2017 *)
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PROG
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(PARI) list(lim)=my(v=List(), t, n); while((t=n++^4+(n+1)^4)<=lim, if(isprime(t) && isprime(n^2+(n+1)^2) && isprime(2*n+1), listput(v, t))); Vec(v) \\ Charles R Greathouse IV, Feb 26 2017
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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