Articles of number theory

Proof of a special case of Fermat's Last Theorem.

Here, I will try to prove a special case of Fermat’s Last Theorem, namely when $a=b$ in this definition: Fermat’s Last Theorem In number theory, Fermat’s Last Theorem (sometimes called Fermat’s conjecture, especially in older texts) states that no three positive integers a, b, and c can satisfy the equation $a^n + b^n = c^n$ […]

prove: $2^n$ is not divisible by 5 for any $n$

there is a prove that there is no $2^n$ that is divisible by 5?

Proving infintely many primes of the form 6k-1

I have seen the past threads but I think I have another proof, though am not entirely convinced. Suppose there are only finitely many primes $p_1, …, p_n$ of the form $6k-1$ and then consider the number $N = (p_1.p_2. … .p_n)^2 – 1$ If all of the primes dividing N are of the form […]

Not a perfect square of the form for any integer x.

Now a days, I become good fan of this site, as this site making me to learn more math..hahaha. Okay! Can we prove that $x^3 + 7$ cannot be perfect square for any positive/negative or odd/even integer of $x$. I checked with number up to x = 1,…1000. I realized that, not a perfect. But, […]

Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$

For any $\ell > 0$ can you find $M, N$ such that $ \boxed{\mathrm{gcd}(x,y) > 1}$ for all $x \in [M, M+\ell]$ and all $y \in [N, N+\ell]$ ? This is related to the statement that the set of integers $(x,y)$ such that $x, y$ are relatively prime has natural density $\frac{6}{\pi^2} \approx \frac{2}{3}$. I […]

Prove or disprove: if $x$ and $y$ are representable as the sum of three squares, then so is $xy$.

Prove or disprove: if $x$ and $y$ are representable as the sum of three squares, then so is $xy$. How to prove or disprove it? I am unable to get any idea on it. It would be of great help if any one could help.

Minimum set of US coins to count each prime number less than 100

Say I wanted to be able to carry enough coins in my pocket such that at any time, I could count out exact change totaling any of the prime numbers less than 100. How would I determine the minimum set of coins I would need to carry? I don’t care about being able to count […]

Asymptotic divisor function / primorials

Let $p_n\#\equiv\prod_{k=1}^{n}p_k$ (primorial), and $\sigma(n)=\sum_{d|n}^{}d$ (divisor function). Could someone please tell me what the general asymptotic of $\dfrac{\sigma(p_n\#)}{p_n\#}$ is? It looks like $O \log(n)$. Is there anything more precise? Update Thank you to Daniel Fischer for his answer. Out of interest, I include the plot of $\dfrac{\sigma(p_n\#)}{p_n\#}$ against $\dfrac{6 e^{\gamma}}{\pi^2} \log p_n$. The primorial curve […]

Is this division theorem already a proven idea?

Ok so my theorem goes like this For any Z = X/Y Where X,Y are positive integers & Y > 0 A1,A2…. are factors of X & B1,B2…. are factors of Y There exists a Bn of Y such that Ai = Bn * Z If Bn < Y then X must be composite If […]

Representations of integers by a binary quadratic form

Let $\mathfrak{F}$ be the set of binary quadratic forms over $\mathbb{Z}$. Let $f(x, y) = ax^2 + bxy + cy^2 \in \mathfrak{F}$. Let $\alpha = \left( \begin{array}{ccc} p & q \\ r & s \end{array} \right)$ be an element of $SL_2(\mathbb{Z})$. We write $f^\alpha(x, y) = f(px + qy, rx + sy)$. Since $(f^\alpha)^\beta$ = […]