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Known primes such that there are no pairwise coprime solutions to the Diophantine equation of the form x^3 + y^3 = p^a z^n with a >= 1 an integer and n >= p^(2p) prime.

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`%I #18 Jul 14 2019 06:28:46
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`%S 53,83,149,167,173,199,223,227,233,263,281,293,311,347,353,359,389,
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`%T 401,419,443,449,461,467,479,487,491,563,569,571,587,599,617,641,643,
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`%U 659,719,727,739,743,751,809,811,823,827,829,839,859,881,887,907,911,929,941,947,953,977,983
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`%N Known primes such that there are no pairwise coprime solutions to the Diophantine equation of the form x^3 + y^3 = p^a z^n with a >= 1 an integer and n >= p^(2p) prime.
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`%C These primes are the prime numbers p greater than 3 such that for every elliptic curves with conductor of the form 18p, 36p, or 72p we have that 4 does not divide the order of the torsion subgroup over the rationals but at least one curve with 2 dividing this order, such that there is a prime q congruent to 1 modulo 6 such that 4 does not divide the order of the torsion subgroup over the finite field of size q.
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`%H M. A. Bennett, F. Luca and J. Mulholland, <a href="http://www.labmath.uqam.ca/~annales/volumes/35-1/PDF/001-015.pdf">Twisted extensions of the cubic case of Fermat's Last Theorem</a>, Ann. Sci. Math. Quebec. 35 (2011), 1-15.
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`%K nonn,obsc
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`%O 1,1
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`%A _Carmen Bruni_, May 15 2012
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