Solutions to a^x + b^x = c^x

Introduction

When considering partial sums and Euler products of zeta(s), we noticed that the solution of the simple equation

2^s + 3^s = 4^s   (originally found as 1/(1 - 1/2^s) = 1 + 1/2^s + 1/3^s)

was not found on the internet. This motivated to consider solutions to other equations of the form

a^x + b^x = c^x   with small positive integers (a, b, c).

When a + b = c, the unique solution is trivially x = 1. We are not interested in that case, nor in Pythagorean triples a² + b² = c².

Equations and solutions

The following table gives a list of the solutions existing or added to OEIS: (lex order of c >= b >= a)

a   b   c   Decimal expansion           Continued fraction
1   2   3   (a + b = c: solution x = 1)
1   2   4   A242208 = 0.69424191363...  A328912 = 0; 1, 2, 3, 1, 2, 3, 2, 4, 2, 1, 2, 11,...     NB: exact value : log_2(Phi = (sqrt(5)+1)/2)
1   3   4   (a + b = c: solution x = 1)
2   3   4   A328900 = 1.507126...       A328913 = 1; ...  
1   2   5   A328905 = 0.563895524...    A329335 = 0; 1, 1, 3, 2, 2, 2, 1, 3, 3, 1, 5, 3, 1, 1, 3,...
1   3   5   A328904 = 0.727160...       A329334 = 0; 1, 2, 1, 1, 1, 72, 1, 3, 2, 6, 1, 1, 2, 45,...
1   4   5   (a + b = c: solution x = 1)
2   4   5       x = 1.42158623076210...       1; 2, 2, 1, 2, 4, 1, 4, 1, 1, 1, 4, 1, 4, 1, 2, 2, 4,...
3   4   5   (Pythagorean triple: x = 2)
1   2   6   A328906 = 0.489536321...  0; 2, 23, 2, 1, 1, 4, 1, 1, 27, 4, 12, 1, 1, 1, 1    
1   3   6   A328907 = 0.60096685...   0; 1, 1, 1, 1, 40, 1, 3, 2, 1, 2, 23, 1, 13, 1, 8, ...

PARI code

s(a,b,c)=solve(x=0,1,a^x+b^x-b^x))
cf(a,b,c)=contfrac(s(a,b,c))
d(a,b,c)=digits(s(a,b,c)\.1^default(realprecision))[^-1]