Leonard Pisano's young daughter, Fifi, was visiting him at work.
Fifi read from a letter on her father's desk. ‘Find a square number which, being increased or diminished by 5, gives a square number. What does that mean?'
'It's John's congruum problem, considered over the rationals,' Leonard replied. 'One solution is 31^2/12^2, 41^2/12^2, and 49^2/12^2, since 49^2/12^2-41^2/12^2=41^2/12^2-31^2=5. It's a neat problem - people will likely still be puzzling over it 800 years from now.'
'And does it just work for a gap of 5?' asked Fifi, 'or other numbers as well?'
'Great question,' said Leonard. 'Solutions for gaps of 6,14,15, and so on are ten a penny. 5 took a little thought, and 7 even more. In principle there should be solutions for 13 and 29, too, but I'm still working on those.'
Can you find a solution with a gap of 7? Can you show that it is minimal?
If so, how about 13 and 29?
For a difference of 7
