Prove That the Following Three Statements About Positive Integers N $n ^ { 3 } + 1$

question 98

Essay

Prove that the following three statements about positive integers n are equivalent: (a) n is even; (b) $n ^ { 3 } + 1$ is odd:
(c) $n ^ { 2 } - 1$ is odd

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Definitions:

Government Guarantees

Government guarantees refer to promises made by the government to back certain obligations of entities, reducing risk for investors and encouraging investment in specific areas.

Socialize Losses

A policy or practice where financial losses are distributed across the broader society, typically through government intervention, instead of being borne solely by those who incurred the losses.

Privatize Gains

The concept where profits are allocated to private individuals or corporations rather than being shared with the public or society at large.

Mohair Production

The process of harvesting and processing the fiber produced by Angora goats, used in textiles.