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Number Theory Level 3

How many integer solutions are there for the equation: $42^x+x^{42}=26^x\cdot x^{26}$ ?

2 solutions

U Z
Jan 3, 2015

$42^x + x^{42} = 26^x .x^{26}$

$26^{2x} + (x^{26})^2 = 2.26^x .x^{26} - 26^x .x^{26}$

$(26^{2x} - x^{26})^2 = -26^x .x^{26}$

Thus zero integral solutions

the solution is not clear ::
how did $42^x$ + $x^{42}$ change into $26^{2x}$ + $(x^{26})^2$

Vaibhav Chowdhry - 6 years ago

$42^x != 26^{2x}$

Vishnu Bhagyanath - 5 years, 11 months ago

Alexander Dimitrakakis
Dec 9, 2015

42^x is different from 26^(2*x)