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Questions tagged [elementary-number-theory]

Questions on divisibility, gcd and lcm, congruences, linear Diophantine equations, Fermat's and Wilson's theorems, the Chinese Remainder theorem, primitive roots, quadratic congruences, quadratic number fields and other related topics which may be treated in first courses on number theory. More advanced topics should instead use the number-theory or other tags.

Questions on introductory topics: divisibility, 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 other related topics which may be treated in first courses on number theory.

For more advanced topics, please use pertinent tags, e.g. , , , or tags.

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Find five positive integers whose reciprocals sum to $1$

Find a positive integer solution $(x,y,z,a,b)$ for which $$\frac{1}{x}+ \frac{1}{y} + \frac{1}{z} + \frac{1}{a} + \frac{1}{b} = 1\;.$$ Is your answer the only solution? If so, show why. I was surprised that a teacher would assign this kind of…
Low Scores
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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…
JDH
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How many fours are needed to represent numbers up to $N$?

The goal of the four fours puzzle is to represent each natural number using four copies of the digit $4$ and common mathematical symbols. For example, $165=\left(\sqrt{4} + \sqrt{\sqrt{{\sqrt{4^{4!}}}}}\right) \div .4$. If we remove the restriction…
David Bevan
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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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Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
user103028
171
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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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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
164
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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 to…
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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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 $a|(c - by)$. $a$, $c$…
Chan
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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 $(a,c)\ne1$, for…
user7530
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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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Am I just not smart enough?

When I was doing math, let us say for example, introductory number theory, it seems to take me a lot of time to fully understand a theorem. By understanding, I mean, both intuitively and also rigorously (know how to prove or derive). However, I…
Kun
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A multiplication algorithm found in a book by Paul Erdős: how does it work?

I am trying to understand the following problem from Erdős and Surányi's Topics in the theory of numbers (Springer), chapter 1 ("Divisibility, the Fundamental Theorem of Number Theory"): We can multiply two (positive integer) numbers together in…
iadvd
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