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A358742
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First of three consecutive primes p,q,r such that p^3 + q^3 - r^3 is prime.
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4
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13, 29, 89, 97, 127, 137, 151, 163, 199, 223, 241, 277, 313, 349, 367, 389, 419, 431, 457, 463, 521, 577, 613, 691, 823, 827, 829, 859, 877, 883, 911, 953, 971, 1049, 1087, 1097, 1129, 1151, 1163, 1217, 1409, 1489, 1499, 1579, 1699, 1723, 1867, 1879, 1993, 2089, 2111, 2141, 2293, 2339, 2399, 2411
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OFFSET
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1,1
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COMMENTS
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Note: for x >= 3275, there is a prime between x and x(1 + 1/(2 log^2 x)) < 1.01x (Dusart 1998). Together with finite checking this shows that for p > 19, p^3 + q^3 - r^3 > 0. - Charles R Greathouse IV, Nov 29 2022
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LINKS
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EXAMPLE
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a(3) = 89 is a term because 89, 97, 101 are consecutive primes and 89^3 + 97^3 - 101^3 = 587341 is prime.
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MAPLE
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R:= NULL: count:= 0: q:= 2: r:= 3:
while count < 100 do
p:= q; q:= r; r:=nextprime(r);
if isprime(p^3+q^3-r^3) then count:= count+1; R:= R, p; fi;
od:
R;
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MATHEMATICA
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Select[Partition[Prime[Range[360]], 3, 1], (s = #[[1]]^3 + #[[2]]^3 - #[[3]]^3) > 0 && PrimeQ[s] &][[;; , 1]] (* Amiram Eldar, Nov 29 2022 *)
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PROG
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(Python)
from itertools import islice
from sympy import isprime, nextprime
def agen():
p, q, r = 2, 3, 5
while True:
if isprime(p**3 + q**3 - r**3): yield p
p, q, r = q, r, nextprime(r)
(PARI) a358742(upto) = {my(p1=2, p2=3); forprime(p3=5, upto, if (isprime (p1^3+p2^3-p3^3), print1(p1, ", ")); p1=p2; p2=p3)};
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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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