### Testing Dujella's unicity conjecture

The conjecture states that the diophantine equation x^{2} – (k^{2} + 1)y^{2} = k^{2} has at most one positive solution (x,y) with 1 ≤ y < k – 1. (For background, see slides.)
This program deals with an equivalent diophantine equation, namely X^{2} – (k^{2} + 1)Y^{2} = -k^{2}. The positive integral solutions (x,y) with 1 ≤ y < k –1 are in 1-1 correspondence with the positive solutions (X,Y) with 1 < Y < k of the equation X^{2} – 1 = (k^{2} + 1)(Y^{2} – 1) under the mapping x = (k^{2} – 1)Y – kX, y = kY – X. (This was observed by John Robertson.)

So Dujella's unicity conjecture is equivalent to there being at most one positive integer solution (X,Y) satisfying 1 < Y < k. In fact we can show that such a solution
must satisfy Y < k/2, which implies 1 < X < (k^{2} + 1)/2.

Accordingly, we look for solutions of the congruence X^{2} ≡ 1 mod(k^{2} + 1) satisfying 1 < X < (k^{2} + 1)/2 and for which (X^{2} – 1)/(k^{2} + 1) = Y^{2} – 1.
Note that if k is odd, then k^{2} + 1 = 2(2m + 1).

We test the conjecture for k in the range m ≤ k ≤ n, where n < 10^{5}, exhibiting the k for which an exceptional solution exists.

*Last modified 23rd November 2014*

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