Prime Numbers

416 Chapter 8 THE UBIQUITY OF PRIME NUMBERS
Observe that in seeking Diophantine solutions here we are simply addressing
the problem of whether there exist higher instances of consecutive powers.
An accessible treatment of the history of the Catalan problem to the date of
its publication is [Ribenboim 1994], while more recent surveys are [Mignotte
2001] and [Metsänkylä 2004]. Using the theory of linear forms of logarithms
of algebraic numbers, R. Tijdeman showed in 1976 that the Catalan equation
has at most finitely many solutions; in fact,
y q 10 7
and max{p, q} < 7.78 × 10 16 . Further, explicit easily checkable criteria on
allowable exponent pairs were known, for example the double Wieferich
condition of Mihăilescu: if p, q are Catalan exponents other than the pair
2, 3, then
p q−1 ≡ 1(modq 2 ) and q p−1 ≡ 1(modp 2 ).
It was hoped that such advances together with sufficiently robust calculations
might finish off the Catalan problem. In fact, the problem was indeed finished
off, but using much more cleverness than computation.
In [Mihăilescu 2004] a complete proof of the Catalan problem is presented,
and yes, 8 and 9 are the only pair of nontrivial consecutive powers. It is
interesting that we still don’t know whether there are infinitely many pairs of
consecutive powers that differ by 2, or any other fixed number larger than 1,
though it is conjectured that there are not. In this regard, see Exercise 8.20.
Related both to Fermat’s last theorem and the Catalan problem is the
Diophantine equation
x p + y q = z r , (8.2)
where x, y, z are positive coprime integers and exponents p, q, r are positive
integers with 1/p+1/q+1/r ≤ 1. The Fermat–Catalan conjecture asserts that
there are at most finitely many such powers x p ,y q ,z r in (8.2). The following
are the only known examples:
1 p +2 3 =3 2
2 5 +7 2 =3 4 ,
13 2 +7 3 =2 9 ,
2 7 +17 3 =71 2 ,
3 5 +11 4 = 122 2 ,
33 8 + 1549034 2 = 15613 3 ,
1414 3 + 2213459 2 =65 7 ,
(p ≥ 7),