Questions tagged [elementary-number-theory]

For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.

For questions on introductory topics in number theory, such as divisibility, prime numbers, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields, Pell's equations, and related topics.

For more advanced topics, use relevant tags, e.g.

39260 questions
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Does $\pi$ contain all possible number combinations?

$\pi$ Pi Pi is an infinite, nonrepeating $($ sic $)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever…
Chani
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257
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What is the importance of the Collatz conjecture?

I have been fascinated by the Collatz problem since I first heard about it in high school. Take any natural number $n$ . If $n$ is even, divide it by $2$ to get $n / 2$ , if $n$ is odd multiply it by $3$ and add $1$ to obtain $3n + 1$ . Repeat the…
Dan Brumleve
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Can you answer my son's fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?

My son Horatio (nine years old, fourth grade) came home with some fun math homework exercises today. One of his problems was the following little question: I am thinking of a number... It is prime. The digits add up to $10.$ It has a $3$ in the tens…
JDH
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1 answer

Are $14$ and $21$ the only "interesting" numbers?

The numbers $14$ and $21$ are quite interesting. The prime factorisation of $14$ is $2\cdot 7$ and the prime factorisation of $14+1$ is $3\cdot 5$ . Note that $3$ is the prime after $2$ and $5$ is the prime before $7$ . Similarly, the prime…
Simon Parker
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210
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7 answers

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
user103028
183
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12 answers

Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$

For all $a, m, n \in \mathbb{Z}^+$, $$\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$$
Juan Liner
178
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6 answers

Deleting any digit yields a prime... is there a name for this?

My son likes his grilled cheese sandwich cut into various numbers, the number depends on his mood. His mother won't indulge his requests, but I often will. Here is the day he wanted 100: But today he wanted the prime 719, which I obliged. When…
Fixee
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166
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4 answers

What happens when we (incorrectly) make improper fractions proper again?

Many folks avoid the "mixed number" notation such as $4\frac{2}{3}$ due to its ambiguity. The example could mean " $4$ and two thirds", i.e. $4+\frac{2}{3}$ , but one may also be tempted to multiply, resulting in $\frac{8}{3}$ . My questions pertain…
Zim
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Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?

I was looking at a list of primes. I noticed that $ \frac{AM (p_1, p_2, \ldots, p_n)}{p_n}$ seemed to converge. This led me to try $ \frac{GM (p_1, p_2, \ldots, p_n)}{p_n}$ which also seemed to converge. I did a quick Excel graph and regression and…
Soham
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Why is $1$ not a prime number?

Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
bryn
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141
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12 answers

Modular exponentiation by hand ($a^b\bmod c$)

How do I efficiently compute $a^b\bmod c$ : When $b$ is huge, for instance $5^{844325}\bmod 21$ ? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, for instance $5^{69}\bmod 101$ ? When $(a,c)\ne1$ ,…
user7530
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5 answers

How to find solutions of linear Diophantine ax + by = c?

I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ? I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$. $a$, $c$ and…
roxrook
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133
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29 answers

Different ways to prove there are infinitely many primes?

This is just a curiosity. I have come across multiple proofs of the fact that there are infinitely many primes, some of them were quite trivial, but some others were really, really fancy. I'll show you what proofs I have and I'd like to know more…
Patrick Da Silva
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127
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16 answers

What is the smallest unknown natural number?

There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact values is also interesting. There are situations…
Joonas Ilmavirta
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124
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15 answers

Infiniteness of non-twin primes.

Well, we all know the twin prime conjecture. There are infinitely many primes $p$, such that $p+2$ is also prime. Well, I actually got asked in a discrete mathematics course, to prove that there are infinitely many primes $p$ such that $p + 2$ is…
Tomas Wolf
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