Find all \(\displaystyle{q}\in{\mathbb{{{N}}}}\) such that \(\displaystyle{\frac{{{q}{\left({q}+{1}\right)}}}{{{12}}}}\)

Liseskirlsojh

Liseskirlsojh

Answered question

2022-04-03

Find all qN such that q(q+1)12 is a Perfect square.

Answer & Explanation

enriuadaziaa

enriuadaziaa

Beginner2022-04-04Added 7 answers

It remains to find r such that 48r2+1=s2 or s248r2=1, which is Pell's equation with fundamental solution s=7,r=1,q=3 as you found. All other solutions are generated by the following recurrence:
[sk+1rk+1]=[74817][skrk]
This may be solved for s=48r2+1, giving
s=(7+43)k+(743)k2,kN
Since we want only positive q, we take the positive sign in the formula for q, yielding q=s12.

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