Exponential Diophantine Equations

1. Historical introduction. Many questions in number theory concern perfect powers, numbers of the form a b where a and b are rational integers with <7>1, 6>1. To mention a few: (a) Is it possible that for /zs>3 the sum of two 77th powers is an /7th power? (b) Is 8, 9 the only pair of perfect powers which differ by 1 ? (c) Is it possible that the product of consecutive integers, (x+l)(x + 2) ... (x+m), is a perfect power? (d) Does a given polynomial with integer coefficients represent (infinitely many) perfect powers at integer points? (e) Can a number with identical digits in the decimal scale be a perfect power? A common feature of these problems is that they can be restated in the form of a diophantine equation in which exponents occur as variables. Problem (a) leads to the equation x"+y n =z n in integers 77^ 3, x>\ 9 y>\ 9 z>l which is still unsolved in spite of Fermat's claim and all efforts thereafter. Problem (b) was posed by Catalan in 1844 and corresponds to the equation x m — y"=l in integers 77/>1, 77>1, x>l, 3^>1. Problem (c) goes back to Liouville who gave a partial solution in 1857. The complete solution was obtained by Erdös and Selfridge in 1975. They showed that /7"=i (*+./)—)'" l ias no solutions in integers m >1, 77>1, x^l 9 y>l. Problem (d) is stated here in a general form, but special cases were investigated long ago. In 1850 V. A. Lebesgue proved that x 2-l-l is never a perfect power for integral x. It follows from a more general result of Legendre in 1798 that x 3 —y s is not a power of 2 for x>y>l. Ramanujan conjectured in 1913 that 2 3 , 2 4 , 2 5 , 2 7 and 2 15 are the only powers of 2 assumed by the polynomial x 2 + l for integral x. In 1948 Nagell proved that the corresponding