Is it possible to find all integers m>0 and n>0 such that n+1∣m2+1 and m+1|n2+1 ?
I succeed to prove there is an infinite number of solutions, but I cannot progress anymore.
Thanks !
Answer
Some further results along the lines of thought of @individ:
Suppose p and s are solutions to the Pell’s equation:
−d⋅p2+s2=1
Then,
m=a⋅p2+b⋅pq+c⋅q2n=a⋅p2−b⋅pq+c⋅q2
are solutions if (a,b,c,d) are: (these are the only sets that I found using the computer)
(10,4,−2,−15)(39,12,−3,−65)
Sadly, the solutions are negative.
Here are some examples:
(m,n)=(−6,−38)(a,b,c,d,p,q)=(10,4,−2,−15,1,4)(m,n)=(−290,−2274)(a,b,c,d,p,q)=(10,4,−2,−15,8,31)(m,n)=(−15171,−64707)(a,b,c,d,p,q)=(39,12,−3,−65,16,129)(m,n)=(−1009692291,−4306907523)(a,b,c,d,p,q)=(39,12,−3,−65,4128,33281)(m,n)=(−67207138138563,−286676378361411)(a,b,c,d,p,q)=(39,12,−3,−65,1065008,8586369)
P.S. I am also very curious how @individ thought of this parametrization.
Attribution
Source : Link , Question Author : uvdose , Answer Author : Yifan Zhu