Prove that discrete math the following statement (if true) or provide a counterexample (if false): For all $n\ge 4,{2}^{n}-1$ is not a prime number.

Jaya Legge
2021-08-10
Answered

Prove that discrete math the following statement (if true) or provide a counterexample (if false): For all $n\ge 4,{2}^{n}-1$ is not a prime number.

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Caren

Answered 2021-08-11
Author has **96** answers

Counter example

n=5 Then

31 is a prime. So statement is false for n=5. As

asked 2021-08-02

Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.

a) the set of sophomores taking discrete mathematics in your school

b) the set of sophomores at your school who are not taking discrete mathematics

c) the set of students at your school who either are sophomores or are taking discrete mathematics

Use these symbols:$\cap \cup$

a) the set of sophomores taking discrete mathematics in your school

b) the set of sophomores at your school who are not taking discrete mathematics

c) the set of students at your school who either are sophomores or are taking discrete mathematics

Use these symbols:

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{ 0, { { 0 } }?

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1 foot is divided into 12 inches. Make a fraction of the distance from 0 to a-d

0 to a. = ___

0 to b. = ___

0 to c. = ___

0 to d. = ___

1 foot is divided into 12 inches. Make a fraction of the distance from 0 to a-d

0 to a. = ___

0 to b. = ___

0 to c. = ___

0 to d. = ___

asked 2022-05-21

Partitions of n where every element of the partition is different from 1 is $p(n)-p(n-1)$

I am trying to prove that p(n| every element in the partition is different of $1)=p(n)-p(n-1)$, and I am quite lost... I have tried first giving a biyection between some sets, trying to draw an example in a Ferrers diagram and working on it... Nevertheless, I have not obtained significant results. Then, I have thought about generating functions; we know that the generating function of $\{p(n){\}}_{n\in \mathbb{N}}$ is $\prod _{i=1}^{\mathrm{\infty}}\frac{1}{1-{x}^{i}}$, so $\{p(n-1){\}}_{n\in \mathbb{N}}$ will have $\prod _{i=1}^{\mathrm{\infty}}\frac{x}{1-{x}^{i}}$ as generating function. So, what we have to prove is that $\prod _{i=1}^{\mathrm{\infty}}\frac{1}{1-{x}^{i}}-\prod _{i=1}^{\mathrm{\infty}}\frac{x}{1-{x}^{i}}=(1-$-x). $\prod _{i=1}^{\mathrm{\infty}}\frac{1}{1-{x}^{i}}$ is the generating function of p(n|every element in the partition is different of 1)... but i'm am not seeing why! Any help or hint will be appreciate it!

I am trying to prove that p(n| every element in the partition is different of $1)=p(n)-p(n-1)$, and I am quite lost... I have tried first giving a biyection between some sets, trying to draw an example in a Ferrers diagram and working on it... Nevertheless, I have not obtained significant results. Then, I have thought about generating functions; we know that the generating function of $\{p(n){\}}_{n\in \mathbb{N}}$ is $\prod _{i=1}^{\mathrm{\infty}}\frac{1}{1-{x}^{i}}$, so $\{p(n-1){\}}_{n\in \mathbb{N}}$ will have $\prod _{i=1}^{\mathrm{\infty}}\frac{x}{1-{x}^{i}}$ as generating function. So, what we have to prove is that $\prod _{i=1}^{\mathrm{\infty}}\frac{1}{1-{x}^{i}}-\prod _{i=1}^{\mathrm{\infty}}\frac{x}{1-{x}^{i}}=(1-$-x). $\prod _{i=1}^{\mathrm{\infty}}\frac{1}{1-{x}^{i}}$ is the generating function of p(n|every element in the partition is different of 1)... but i'm am not seeing why! Any help or hint will be appreciate it!

asked 2021-11-10

Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics