Articles of elementary number theory

Integer ordered pairs $(x,y)$ for which $x^2-y!$…

[1] Total no. of Integer ordered pairs $(x,y)$ for which $x^2-y! = 2001$ [2] Total no. of Integer ordered pairs $(x,y)$ for which $x^2-y! = 2013$ My Try:: (1) $x^2-y! = 2001\Rightarrow x^2 = 2001+y!$ We Know that $y!$ end with $0, 1,2,4,6$ and last digit of $x^2$ is $0,1,4,5,6,9$ But I Did Not understand […]

Prove a property of divisor function

Let $n$ be a positive natural number whose prime factorization is $n=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$, where $p_i$ are natural distinct prime numbers, and $a_i$ are positive natural numbers. Using induction to show that the number of divisors of $n$ is $$(a_1+1)(a_2+1)\cdots(a_k+1)$$

$\forall m\in\mathbb N\exists n>m+1\exists N\in\mathbb N:m^2+n^2+(mn)^2=N^2$

Prove the conjecture or give a counter-example: For each $m\in\mathbb N$ there exist a $n>m+1$ such that $m^2+n^2+(mn)^2$ is a perfect square. I have just tried it out numerically and it holds for $m<1000$. I can’t see any pattern for the smallest $n$: m n 1 12 2 8 3 18 4 32 5 50 […]

How many fractional digits do I need to represent a number of base $m$ in base $n$?

I have authored a web application RADIX, which for further optimisation needs to calculate the maximum number of places necessary to precisely represent the fraction part of a number of base $m$ in base $n$ (assuming a precise conversion is possible). For example, assuming $f$ represents $15$ in base $64$, how many fraction digits are […]

Prove that if m is prime and m|kl then either m|k or m|l.

Proofs homework question, here’s what I’ve figured out thus far. Suppose m doesn’t divide k. We need to then prove that m|l. If m doesn’t divide k and m is a prime then we know m and k are co-prime – hcf (m,k) = 1. Which means 1 = ks + mt (for some integers; […]

Solve $V_1+V_2+\cdots+V_k=A, V_1^2+V_2^2+\cdots+V_k^2=B$ in positive integers

There have been changes made to the second equation in the pair that will be worth looking at. All values for the solutions must be non-zero positive integers (natural numbers). Please note, all values must be distinct! According to this equation, $$V_1^2+V_2^2+\cdots+V_k^2 = \left(\frac{2\left(V_{1}+V_{2}+\cdots+V_{k}\right)}{k}-V_{1}\right)^{2}+\left(\frac{2\left(V_{1}+V_{2}+\cdots+V_{k}\right)}{k}-V_{2}\right)^2+\cdots+\left(\frac{2\left(V_{1}+V_{2}\cdots+V_{k}\right)}{k}-V_{k}\right)^{2}$$ (Source: Personal observation) There can be at most only two common […]

Prove that $\nu(n) \le \nu(2^{n}-1)$ where $\nu(n)$ is the number of positive divisors of n

This question already has an answer here: Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$. 1 answer

System of equations – modular arithmetic

I am asked to solve the following….. Let $n\in \mathbb{N}$ and suppose that $a,b,c,d,k,l\in\mathbb{Z}$. Consider the system $ax + by \equiv k$ mod $n$ and $cx+dy \equiv l$ mod $n$. Let $D=ad-bc$. Show that if $hcf(D,n)=1$ then the system has a solution. There were previous parts to the question, in which I showed $ax\equiv b$ […]

Repeatedly multiplying a number by $10^k+1$, where the $k$ rightmost digits are nonzero

Let $n$ be an integer not ending with zero in its decimal representation. Then there are $k \geq 1$ nonzero digits in $n$ (starting from the right side until we hit a zero). Let’s multiply $n$ by $10^k+1$ and do the same procedure for the resulting number. For example, given $n=102035$, we see $k=2$ and […]

Solving a congruence with high numbers and computing the Legendre symbol

I am asked to compute $\left(\frac{77}{257}\right)$ specifically using Euler’s criterion. I have manged to get the following: $$\left(\frac{77}{257}\right)\equiv77^{256/2} \bmod257$$ $$\equiv77^{128}\bmod257$$ I don’t know where to go from here as I don’t know the trick to simplify congruences that are big like this without typing it in on an online calculator. I believe it may be […]