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 $a^n + b^n = c^n$ […]

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

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 […]

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, […]

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 $xy$. 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.

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 […]

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 […]

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 […]

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 $(f^\alpha)^\beta$ = […]

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