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A281792
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Primes of the form x^2 + p^4 where x > 0 and p is prime.
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2
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17, 41, 97, 137, 181, 241, 277, 337, 457, 641, 661, 757, 769, 821, 857, 881, 977, 1109, 1201, 1237, 1301, 1409, 1697, 2017, 2069, 2389, 2417, 2437, 2617, 2657, 2741, 2801, 3041, 3217, 3301, 3329, 3541, 3557, 3697, 3761, 3989, 4001, 4177, 4241, 4337, 4517, 4721, 5557, 5641, 5857, 6101, 6257, 6481, 6577
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OFFSET
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1,1
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COMMENTS
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Heath-Brown and Li prove an asymptotic formula for the number of terms <= x, in particular showing that the sequence is infinite.
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LINKS
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D.R. Heath-Brown and X. Li, Prime values of a^2+p^4, arXiv:1504.00531 [math.NT], 2015; Inventiones Mathematicae page 1-59 (Sep 30 2016), doi:10.1007/s00222-016-0694-0
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FORMULA
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Heath-Brown and Li prove that there are c*x^(3/4)/log^2 x terms up to x, where c = 4*nu*J = 4.79946121442200811438003177..., nu = A199401, and J = A225119. - Charles R Greathouse IV, Aug 21 2017
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EXAMPLE
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17 = 1^2 + 2^4
41 = 5^2 + 2^4
97 = 9^2 + 2^4
137 = 11^2 + 2^4
181 = 10^2 + 3^4
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MAPLE
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N:= 10000: # to get all terms <= N
A:= select(isprime, {seq(seq(x^2+y^4, x=1..floor(sqrt(N-y^4))),
y=select(isprime, [$1..floor(N^(1/4))]))}):
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MATHEMATICA
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nn = 10000;
Select[Table[x^2+y^4, {y, Select[Range[nn^(1/4)], PrimeQ]}, {x, Sqrt[nn-y^4 ]}] // Flatten, PrimeQ] // Union (* Jean-François Alcover, Sep 18 2018, after Robert Israel *)
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PROG
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(PARI) list(lim)=if(lim<17, return([])); my(v=List(), p4, t); forstep(a=1, sqrtint(-16+lim\=1), 2, if(isprime(t=a^2+16), listput(v, t))); forprime(p=3, sqrtnint(lim-4, 4), p4=p^4; forstep(a=2, sqrtint(lim-p4), 2, if(isprime(t=p4+a^2), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 13 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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STATUS
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approved
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