[seqfan] Re: quintic diophantine q^2=2*t^5-1
Walter Kehowski
wkehowski at cox.net
Tue Aug 20 11:04:55 CEST 2019
Peter Crescenzo, A Diophantine Equation Which Arises in the Theory of Groups, Advances in Mathematics 17, 25-29 (1975).
> On August 20, 2019 at 1:51 AM "Richard J. Mathar" <mathar at mpia-hd.mpg.de mailto:mathar at mpia-hd.mpg.de > wrote:
>
>
> 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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