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In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only nontrivial integer solutions of the exponential Diophantine equation (x ^{m  1})/(x1)= (y ^{n  1})/(y  1) satisfying x, y > 1 and n, m > 2 are * (x, y, m, n) = (5, 2, 3, 5); and This may be expressed as saying that 31 and 8191 are the only two numbers which are repunits with at least 3 digits in two different bases. Balasubramanian and Shorey have proved that there are only finitely many possible solutions to the equations in (x,y,m,n) with prime divisors of x and y lying in a given finite set and that they may be effectively computed. See also * FeitThompson conjecture
* R. Balasubramanian; T.N. Shorey (1980). "On the equation a(xm1)/(x1) = b(yn1)/(y1)". Math. Scand. 46: 177–182.
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