Articles of number theory

How to prove Pell's equation for a specific case?

My problem is proving $x^2 – 13y^2 = 1$ has integers solutions. I can find easily see that (+-1, 0) are trivial solution. My question is: is it sufficient to complete the proof? How can I approach this problem? Thanks, Chan

Sum of reciprocals of primes for known primes.

I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges: $\sum_{\text{known primes}}\frac{1}{p} < 4$. However these notes were written pre-$2003$. I was wondering if this is still the case. If not then how much bigger has this sum got?

Show that a prime divisor to $x^4-x^2+1$ has to satisfy $p=1 \pmod{12}$

Suppose that $x$ solves $x^4-x^2+1= 0 \pmod p$. Show that $p=1 \pmod {12}$. Following a hint I have rewritten the equation as $(x^2-1)^2=-3 \mod p$ and $(2x^2-1)^2=-x^2 \pmod p$. The first equation gives that $p=1 \pmod 3$ by using quadratic reciprocity and noting that $1$ is the only applicable quadratic residue. However, I am not […]

Minimal modulus for the finite field NTT

I need your support. Suppose I am performing an NTT in a finite field $GF(p)$. I assume it contains the needed primitive root of unity. I am using it to compute the convolution of two vectors of length $n=2^m, m\in \mathbb{N}$. As usual, I double the length of the vectors to $2n=2^{m+1}$ to make the […]

Is a factorial-primorial mesh ever divisible by the primorial?

In a recent question, I defined a function $n!\#$ as follows: $$n!\# = \prod_{i=1}^n (p_i + i) = (2 + 1)(3 + 2)(5 + 3)(7 + 4) \dots (p_n + n)$$ Modified Question In light of such a trivial solution to my original question (see below), I am going to ask something a bit more […]

sum of floor (“weighted” divisor summary function)

is there a closed form for $$ \sum_{i=1}^N\left\lfloor\frac{N}{i}\right\rfloor i $$ Or is there any faster way to calculate this for any given N value?

How to calculate number of possible voting outcomes

Problem Suppose, we have a voting with n voters and m candidates. Each voter can vote for 1 candidate. How many possible outcomes of voting can be? Solution I tried For 1 candidate there will always be 1 outcome. Possible outcomes for 2 candidates: 1 voter 2 voters 3 voters 4 voters |1|0| |2|1|0| |3|2|1|0| […]

Geometric Interpretation of Complex Algebraic Proof of Sum of Squares Statement

When I see answers regarding proofs such as the one mentioned here, it seems that there is a considerable diversity of ways to attempt to look at this proof. Similarly, although this sum of squares question is very different, I was wondering if there was a geometric interpretation to the following problem: Prove that given […]

systems of congruences and CRT

I want to establish an efficient method to solve linear congruences to prove the Chinese Remainder Theorem. I need a proper generalization/proof for the following ones to go to further developments. If $a$ and $b$ are integers, then $a \bmod b = {a, a-b, a+b, a-2b, a+ 2b,…}$ Let $a_1, a_2, … , a_n, b$ […]

What is linear, numerically and geometrically speaking?

For as simple as it is, I never fully grasped what mathematicians and physicists mean with linear . Intuitively anything that looks like a straight line is interpreted as linear, like something in the form $f(x) = mx + q$ or any other function that maps $R \rightarrow R$ resulting in a graph that looks […]