Solution Verification and Analysis of the Equation ( frac{a^2 b^2 - 1}{ab} in mathbb{N} ) in Number Theory

This article provides a detailed exploration of the problem of finding pairs ( (a, b) ) in ( mathbb{N}^2 ) such that ( frac{a^2 b^2 - 1}{ab} ) is a natural number. The primary focus is on both the verification of known solutions and the analysis of the conditions and constraints that govern the pairs.

Introduction

The given equation is ( frac{a^2 b^2 - 1}{ab} in mathbb{N} ), which we refer to as ( frac{a^2 b^2 - 1}{ab} k ) where ( k ) is a natural number. We aim to find all pairs of natural numbers ( (a, b) ) satisfying this condition.

Initial Observations

To begin with, it is clear that ( a b ) is not a valid solution because ( 2a ) is even, and hence it cannot divide ( 2a^2 - 1 ), which is odd. Therefore, we can assume that ( ab geq 0 ), and assuming ( ab 0 ), the only solution is ( a 0, b 1 ) or ( a 1, b 0 ). For all other cases, we have ( ab > 0 ).

Analysis of Odd and Even Pairs

When ( a 2n - 1 ) is odd, for the equation to hold, ( b ) must also be even. This is because ( ab ) is even, and hence ( ab ) cannot divide ( a^2 b^2 - 1 ), which is odd. Therefore, if ( b ) is even, we can write ( b 2k ) where ( k geq n - 1 ). This gives us ( ab 2n - 2k - 1 ) and ( a^2 b^2 - 1 (2n - 1)^2 (2k)^2 - 1 4n^2 - 4nk 4k^2 - 1 ).

The specific form of ( ab ) and ( a^2 b^2 - 1 ) allows us to simplify the equation to:
( ab - 2a^2 - 1 2n - 2k - 1 - 8n^2 8nk - 8k^2 1 0 )
( b 8n^2 - 6n 2k ) where ( k geq n - 1 ).

The condition for ( b ) to be a valid solution is ( 2n - 2k - 1 8n^2 - 8nk 8k^2 - 1 ) which simplifies to ( b 8n^2 - 6n 2k ).

Other solutions can exist corresponding to ( b d - 2n - 1 ) where ( d ) is a divisor of ( 8n^2 - 8nk 8k^2 - 1 ) greater than ( 4n^2 - 2n^2 k^2 ).

Even 'a' Case

When ( a 2n ) is even, ( b 2k 1 ) must be odd. This again leads to the form ( ab 2n - 2k - 1 ) and ( a^2 b^2 - 1 8n^2 - 8nk 8k^2 - 1 ). The simplification here leads to ( b 8n^2 - 2n - 1 ) as the primary solution. Other solutions exist if ( 8n^2 - 1 ) is not prime and can be expressed as ( b d - 2n ) where ( d ) is a divisor of ( 8n^2 - 1 ) greater than ( 4n ).

Conclusion and Further Observations

In summary, for given ( a ), we can derive corresponding ( b ) values that satisfy the equation ( frac{a^2 b^2 - 1}{ab} k ) where ( k ) is a natural number. The specific forms of ( b ) generated depend on whether ( a ) is odd or even, and they are derived through algebraic manipulations and divisibility conditions.

The solutions for small values of ( a ) are: ( (a, b) (0, 1), (1, 2), (1, 2), (2, 7), (3, 16), (4, 7), (5, 29), (6, 12) ... )