Please help devise a formula that will solve x^2 ┴ (x ┴ 7)^2 = c^2 , where c is a whole number. Thank you.
also 65 and 97... but I need a formula, please.I've found (by brute force) 1, 7, 13, 17, 35, 103
Perhaps...4. Probably there is an unlimited number of values which will satisfy the equation.
I was thinking we could use the rule
but I got lost trying to figure it out.
There is a page of information here about the infinite sequence of solutions of the Diophantine equation . Those solutions obviously satisfy the equation , where If you then put , you will have an infinite sequence of solutions of the equation
I think that in that way you will get all the solutions for which x is a multiple of 7. But there are also plenty of other solutions.
Pell numbers Those numbers don't look as though they will be integers, but when you expand the binomial series for , the terms with all cancel out, leaving you with an integer. The first few such integers are When you form , you get the numbers In each case, you can check that is a square. As I said in my previous comment, you don't get anything like all the possible values of in that way. But I think that you do get all those that are multiples of 7.