About the irrationality of π + e \pi+e and π e \pi e

Number Theory Level 2

π \pi and e e are the two most famous irrational constants in Mathematics.

True or False?

It is certain that among π + e \pi+e and π e \pi e , at least one is irrational.

False True

Victor Dumbrava by Victor Dumbrava , 17, Romania

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1 solution

Suppose π \pi and e e are the roots of a (monic) quadratic, i.e., ( x − π ) ( x − e ) = x 2 − ( π + e ) x + π e = 0 (x - \pi)(x - e) = x^{2} - (\pi + e)x + \pi e = 0 .

Since π \pi and e e are transcendental, i.e., are not algebraic, we know that this quadratic must have at least one irrational coefficient. This implies that at least one of π + e \pi + e or π e \pi e is irrational.

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