[seqfan] Re: quintic diophantine q^2=2*t^5-1

[Walter Kehowski]

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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