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A028916
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Friedlander-Iwaniec primes: Primes of form a^2 + b^4.
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38
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2, 5, 17, 37, 41, 97, 101, 137, 181, 197, 241, 257, 277, 281, 337, 401, 457, 577, 617, 641, 661, 677, 757, 769, 821, 857, 881, 977, 1097, 1109, 1201, 1217, 1237, 1297, 1301, 1321, 1409, 1481, 1601, 1657, 1697, 1777, 2017, 2069, 2137, 2281, 2389, 2417, 2437
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OFFSET
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1,1
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COMMENTS
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John Friedlander and Henryk Iwaniec proved that there are infinitely many such primes.
Primes of the form (x^2 + y^2)/2, where x > y > 0, such that (x-y)/2 or (x+y)/2 is square. - Thomas Ordowski, Dec 04 2017
Named after the Canadian mathematician John Benjamin Friedlander (b. 1941) and the Polish-American mathematician Henryk Iwaniec (b. 1947). - Amiram Eldar, Jun 19 2021
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LINKS
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EXAMPLE
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2 = 1^2 + 1^4.
5 = 2^2 + 1^4.
17 = 4^2 + 1^4 = 1^2 + 2^4.
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MAPLE
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N:= 10^5: # to get all terms <= N
S:= {seq(seq(a^2+b^4, a = 1 .. floor((N-b^4)^(1/2))), b=1..floor(N^(1/4)))}:
sort(convert(select(isprime, S), list)); # Robert Israel, Oct 02 2015
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MATHEMATICA
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nn = 10000; t = {}; Do[n = a^2 + b^4; If[n <= nn && PrimeQ[n], AppendTo[t, n]], {a, Sqrt[nn]}, {b, nn^(1/4)}]; Union[t] (* T. D. Noe, Aug 06 2012 *)
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PROG
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(PARI) list(lim)=my(v=List([2]), t); for(a=1, sqrt(lim\=1), forstep(b=a%2+1, sqrtint(sqrtint(lim-a^2)), 2, t=a^2+b^4; if(isprime(t), listput(v, t)))); vecsort(Vec(v), , 8) \\ Charles R Greathouse IV, Jun 12 2013
(Haskell)
a028916 n = a028916_list !! (n-1)
a028916_list = map a000040 $ filter ((> 0) . a256852) [1..]
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CROSSREFS
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Primes of form n^2 + b^4, b fixed: A002496 (b = 1), A243451 (b = 2), A256775 (b = 3), A256776 (b = 4), A256777 (b = 5), A256834 (b = 6), A256835 (b = 7), A256836 (b = 8), A256837 (b = 9), A256838 (b = 10), A256839 (b = 11), A256840 (b = 12), A256841 (b = 13).
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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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