Articles of number theory

Comparing sums of reciprocals

Prove (or disprove) the following statement: For any positive integers $x,y,t$, $\displaystyle\sum_{i=1}^{t(y+1)-1} \frac{1}{t(xy+x-1)-x+i}$ is an increasing function of $t$. My attempts: The statement appears to be true numerically. Tried some obvious bounds to compare the sums for consecutive values of $t$ but didn’t find one that was strong enough to prove the statement.

Are there useful criterions whether a positive integer is the difference ot two positive cubes?

This question A Diophantine equation involving factorial made me try to find a useful structure for the set of positive integers that are the difference of two cubes. I have two questions similar to the linked question : Define $S$ to be the set of positive integers $N$, that are the difference of two positive […]

Determine the last two digits of $3^{3^{100}}$

Determine the last two digits of $3^{3^{100}}$ This is one of the problems in the past exam my modern algebra course. I think I need to use euler-fermat theorem but can’t figure out how to use it for this problem. Can anyone help me out?

For which primes $p$ does $px^2-2y^2=1$ have a solution?

Let $p$ be an odd prime. If $px^2-2y^2=1$ is solvable, we can get Jacobi symbol $(\frac{-2}{p})=1$, so $p=8k+1,8k+3$. But when $k=12$, $p=97$, the Pell equation $97x^2-2y^2=1$ is unsolvable. I think this diophantine equation is unsolvable for many integers. But it satisfies the Jacobi symbol, so my question is for what prime $p$, the diophantine equation […]

Fundamental unit in the ring of integers $\mathbb Z$

Find a fundamental unit in the ring of integers $\mathbb Z[\frac{1+\sqrt{141}}{2}]$ of $\mathbb Q(\sqrt{141})$ I have different corollaries for different numbers, the most appropriate for $141$ is the one below. I used an algorithm (don’t know if you know this, but $\beta_0=\sqrt{141}+\lfloor\sqrt{141}\rfloor, \quad\beta_{n+1}=\frac{1}{\beta_n-\lfloor\beta_n\rfloor}$ $a_n=\lfloor\beta_n\rfloor$ $p_n=p_{n-1}a_n+p_{n-2}, \quad q_n=q_{n-1}a_n+q_{n-2} $) to determine the continued fraction expansion of […]

How to solve $x^3\equiv 10 \pmod{990}$?

How to solve $x^3\equiv 10\pmod{990}$? It has 3 solutions: 10, 340, 670 (WolframAlpha).

Contradiction: Prove 2+2 = 5

This question already has an answer here: $2+2 = 5$? error in proof 7 answers

Is $\log(n!) \in\Theta(n \log n)$

This question already has an answer here: Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. 8 answers Proving Asymptotic Barrier – O notation [duplicate] 1 answer

Solving a Diophantine Equation

Let $p \equiv q \equiv 3 \pmod 4$ for distinct odd primes $p$ and $q$. Show that $x^2 – qy^2 = p$ has no integer solutions $x,y$. My solution is as follows. Firstly we know that as $p \equiv q \pmod 4$ then $\big(\frac{p}{q}\big) = -\big(\frac{q}{p}\big)$ Assume that a solution $(x,y)$ does exist and reduce […]

Property of Derivative in a local field

Let $K$ be a charateristic $p$ global field. For each place $P$ we let $K_P$ to be the completion at $P$ (which is a local field) and let $t_P$ be their prime. Let $t$ be a separating element of $K$ (which means $K/F(t)$ is a separable extension where $F$ is the constant field over $K$). […]