(a) Given an integer x greater than 2 such that x3−x2+ 1 is even, prove that the x-th power of the x-th prime is always odd. (b) Prove that if x is a positive integer such that x4/log(x) > 3pln(x), then x3+ x > x2−x.

(a) Given an integer x greater than 2 such that x3−x2+ 1 is even, prove that the x-th power of the x-th prime is always odd. (b) Prove that if x is a positive integer such that x4/log(x) > 3pln(x), then x3+ x > x2−x.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

For each of the following conjectures, determine what method of proof
would be the most efficient method for proving the statement. You need
not write a formal proof (though you can) but you must give an expla-
nation for why the method you chose would be most efficient.
(a) Given an integer x greater than 2 such that x3−x2+ 1 is even, prove
that the x-th power of the x-th prime is always odd.
(b) Prove that if x is a positive integer such that x4/log(x) > 3pln(x),
then x3+ x > x2−x.

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