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Does it continue? 9: Diophantine friends (Posted on 2017-10-03) Difficulty: 3 of 5
Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Does each of the two Diophantine equations have just the exact same five positive solutions of x = {1,2,3,6,91}?

2x2(x2-1) = 3(y2-1)
x(x-1)/2 = 2z-1

No Solution Yet Submitted by Jer    
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Some Thoughts Thoughts, no proof. Comment 1 of 1

These look very much like Pell Equations and also elliptic curves.

A limited number of solutions is therefore expected, particularly with square x in the first equation, for which we could write 2a(a-1) = 3(y2-1). Note that we would then have (a-sqrta)= 2(2z-1) in the second equation.

Therefore I expect it is true.

Edited on October 4, 2017, 12:34 am
  Posted by broll on 2017-10-04 00:29:57

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