Order between symmetric powers on positive integers
DOI:
https://doi.org/10.35819/remat2023v9i2id6536Keywords:
Order, Induction, Integers Numbers, Symmetric PowersAbstract
It is common to see mathematical challenges like: what is the greater value, 20^(33) or 33^(20)? Encouraged by this type of problem of comparison between symmetrical powers, in this article we will demonstrate that, for any x and y that are positive integers, with y>x>1, the inequality (x^y)>(y^x) holds, except for the pairs y=3, x=2 and y=4, x=2. That is, with these two exceptions, the power x^y of the larger exponent is greater than the power y^x of the larger base. For this we will use the principle of induction, the elementary derivatives and the fundamental exponential limit.
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References
FIGUEIREDO, Djairo G. Análise I. Rio de Janeiro: LTC, 1974.
THOMAS, George B. Cálculo 1. v. 1. 10. ed. São Paulo: Addison Wesley, 2002.
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