0
votes
1answer
40 views

Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
2
votes
0answers
191 views

The logarithm of a product

Let $p$ be a prime number, $C\in \mathbb{N}$ and C is not a square. Then define $$F=\prod_{|z| \leq \sqrt{\frac{x}{2}} \atop |y|\leq \sqrt{\frac{x}{2D}}}{|z^2-Cy^2|}.$$ Note that we omit the term with ...
15
votes
1answer
526 views

Prove that the product of some numbers between perfect squares is $2k^2$

Here's a question I've recently come up with: Prove that for every natural $x$, we can find arbitrary number of integers in the interval $[x^2,(x+1)^2]$ so that their product is in the form of ...
1
vote
0answers
30 views

Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
4
votes
1answer
93 views

Distribution of Digit Products

A digit product $P(n)$ of a natural number $n$ is given by the product of its decimal digits. For example: $$P(1234) = 24,\;\;\; P(24) = 8,\;\;\; P(8) = 8$$ $$1\times2\times3\times4 = 24, \;\;\; ...
4
votes
2answers
184 views

Are Euclid numbers squarefree?

Are Euclid numbers squarefree ? An Euclid number is by definition a Primorial number + 1. See http://mathworld.wolfram.com/Primorial.html. In notation the $n$ th Euclid number is written as $E_n = ...
2
votes
3answers
1k views

About the factors of the product of prime numbers

If a number is a product of unique prime numbers, are the factors of this number the used unique prime numbers ONLY? Example: 6 = 2 x 3, 15 = 3 x 5. But I don't know for large numbers. I will be using ...