[Guest]

Determine all possible pair(s) (m, n) of positive integers such that (n² + 1)/(mn – 1) is an integer.

[HoustonHokie]

pairs of (m,n) that fit:

1,2 => 5

1,3 => 5

2,1 => 2

2,3 => 2

3,1 => 1

3,2 => 1

I think that's all

Wow - it's been a while since I've answered anything here.

[Guest]

Algebraically

let the number formed by the expression be "k"

n²+1=kmn-k

n²-kmn=1-k

n(n-km)=1-k

(and since k is greater than or equal to 1, it'll be more helpful to multiply by -1)

n(km-n)=k-1

taking that in modulo n

0=k-1 (mod n)

1=k (mod n)

so k=1, 1+n, 1+2n, 1+3n....

Case 1

If k=1 then,

n²+1=mn-1

n(m-n)=2

If n=1, then m=3

If n=2, then m=3

Case 2

If k=1+n

n²+1=(1+n)(mn-1)

n²+1=mn²+mn-n-1

2=²(m-1)+n(m-1)

2=n(n+1)(m-1)

So if n=1, then m=3

n cannot be any other number because of the n(n+1) term

So if we continue with each case and use k=1+2n and k=1+3n and so on, we will eventually find all of the values.