[seqfan] quintic diophantine q^2=2*t^5-1
Richard J. Mathar
mathar at mpia-hd.mpg.de
Tue Aug 20 10:51:25 CEST 2019
In conjunction with A161460 (pairs of tau-values that repeat
in the list of the number-of-divisors function) one may ask
whether 50175 is in the sequence, and one of the subtasks
seems to be to find pairs of (q,t) such that q^2=2*t^5-1.
Apparently this inhomogeneous quintic diophantine equation has only the
solution q=t=1. Is there any proof to that?
Searching for odd q and t with a substitution q=2*q'+1, t=2*t'+1
does not lead far,
q'*(q'+1)= t'*(16*t'^5 + 40 t'^4+ 40 t'^3 + 20 t'^2 + 5'*t')
because the usual unique factorization methods are not applicable:
q' and t' are not prime.
For comparison: the quadratic diophantine Pell equation q^2=2*t^2-1
has q=1,7,41,239,1393,8119..
t=1,5,29,169,985,..
as in A002315 and A001653.
The cubic q^2=2*t^3-1 seems to have only the "trivial"
solution q=t=1.
The quartic q^2=2*t^4-1 has (obviously) a subset of the
quadratic solutions if we sieve for perfect squares in A001653.
This yields at least two solutions at
q=1,239,..
t^2=1,169,..
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