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A282867
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Primes of the form x^2 + y^2 with x > y such that x^2 - y^2 is a square and x^4 + y^4 is a prime.
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1
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41, 313, 3593, 4481, 32633, 42961, 66361, 67073, 165233, 198593, 237161, 266921, 378953, 462073, 465041, 487073, 559001, 594161, 750353, 757633, 815401, 1157033, 1414081, 1416161, 1687393, 2439881, 2793481, 2866121, 2947561, 3344161, 3577913, 3759713, 4295281, 4617073, 4795481, 5654641
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OFFSET
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1,1
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COMMENTS
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Primes of the form (u^4 + v^4)/2 with u and v odd and (u^8 + 6*u^4*v^4 + v^8)/8 prime. - Robert Israel, Feb 24 2017
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LINKS
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FORMULA
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a(n) == 1 (mod 8).
a(n) == 1 or 33 (mod 40).
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EXAMPLE
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For prime 41 = 5^2 + 4^2 is 5^2 - 4^2 = 3^2 and 5^4 + 4^4 = 881 is prime.
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MAPLE
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N:= 10^7: # to get all terms <= N Res:= {}:
for w from 1 to floor((2*N)^(1/4)) by 2 do
for u from 1 to min(w-1, floor((2*N-w^4)^(1/4))) by 2 do
p:= (u^4 + w^4)/2;
if isprime(p) and isprime((u^8 + 6*u^4*w^4 + w^8)/8) then
Res:= Res union {p}
fi;
od od:
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MATHEMATICA
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Select[Total[#^2]&/@Select[Subsets[Range[3000], {2}], IntegerQ[Sqrt[#[[2]]^2-#[[1]]^2]] && PrimeQ[ Total[#^4]]&], PrimeQ]//Union (* Harvey P. Dale, Jul 23 2024 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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