Why must N be odd in Exercise 3.16?

Name: Problem 17, Chapter 3: Digital Design and Computer Architecture: RISC-V
Uploaded: 2021-06-11T16:51:32-08:00
Duration: 3 min 8 s
Channel: Nick Johnson
Description: Why must N be odd in Exercise 3.16?

Key Concepts

Integer Parity

Parity distinguishes between odd and even integers, a fundamental concept in number theory and combinatorics. In many mathematical constructions, knowing whether a number is odd or even can affect properties such as symmetry, the existence of fixed points, or cancellation effects, which in turn determine whether certain configurations are possible.

Symmetry and Cancellation

When designing or analyzing mathematical structures like sequences, matrices, or geometrical arrangements, symmetry plays a critical role. With an even number, symmetric pairings may lead to unwanted cancellations or duplicate contributions, while an odd number can avoid such issues and ensure that the structure exhibits the intended non?degenerate behavior.

Constraints for Existence of Constructions

Certain existence proofs or constructions in mathematics require parameter values to satisfy specific criteria. The need for a parameter to be odd may stem from the requirement that the configuration maintains a non-trivial structure, as an even parameter might force a degeneracy or symmetry that obstructs the desired properties.

Transcript

00:01 Okay, so here we want to prove that n is odd if and only if, n squared is odd.

00:09 Well, in problem 73, we already proved that if n is odd, then n squared is odd, and that if n squared is odd, then n is odd.

00:18 So that is going to prove both directions.

00:20 Prove an if and only if statement.

00:22 We have to prove both directions, which, well, we already did in the preceding statement.

00:27 So again, it's going over time.

00:28 We show that, well, if n is odd, then what do we have? if n is odd, then we had that n is equal to 2k plus 1 for some integer k.

00:43 So plugging that in for n squared, we said n squared is equal to 2k plus 1 squared.

00:49 So therefore, n squared is equal to 2k plus 1 times 2k plus 1 is a 4k squared plus 4k plus 1.

00:55 Or the 4k squared plus 4k, we can factor out a 2 from that.

00:59 So that's going to be equal to 2 times 2k squared plus 2k, but 2k squared plus 2k is just 2 times an integer plus an integer, that's just going to be an integer.

01:09 Let's call that integer j.

01:12 So therefore n squared is equal to 2j plus 1, so therefore n squared is equal to two times an integer plus 1, which is odd.

01:21 So therefore we show that at one direction, right, if n is odd, then n squared is odd.

01:27 And then to show the other direction, to show that if n squared is odd, then n is odd, or then we proved the contrapositive.

01:35 We took the contrapositive of our statement...

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