# Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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### Question on number theory [closed]

If a²+(5/2)b²+c²= 2ab+bc+ca then a+2b+2c equals what? I am unable to think how to solve it.. Please help
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### Number of even numbers having digit 2 in them.

I am trying to count numbers from 1 to N which exist in A121022 but I am unable to think of solving in better than O(NLog(N)) , can you suggest a better algorithm?
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### Some questions on Euler's phi function

I was reading Number Theory by George E. Andrews (Dover 1994,) problem set 6-1, p. 81. (I'm not a student; I just find problems like these entertaining like some people enjoy crosswords or Sudoku.) ...
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### Can an irrational number be expressed as a sum of other irrational numbers, at least one of which is not an integral multiple of the required number?

For example, $\pi = Ae + B\sqrt 2+ \cdots$ ($A,B,\ldots\in\mathbb R$) (Equations like "$\pi = 3\pi - 2\pi$" are not allowed.)
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### How can we create arbitrarily long instances of the Euclidean algorithm?

How can we create arbitrarily long instances of the Euclidean algorithm? What kind of numbers are useful? What is the relationship between the size of these numbers and the number of steps?
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### Why does the Euclidean algorithm always terminate?

Why does the Euclidean algorithm always terminate? Can we make this effective by bounding the number of steps it takes in terms of the original integers?
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### Is $n^7 - 77$ ever a Fibonacci number?

As the question title suggests, is $n^7 - 77$ ever a Fibonacci number, where $n$ is a integer?
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### Given the value of a polynomial mod $611953$, find $x$?

Given a polynomial of degree $n$, and a value $\pmod{611953}$, find the possible $x$ at which this value occurs? For example a polynomial $p(x)$ is given of some degree $n$, and a number is given ...
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### Existence of elliptic curves for thriples a,b,c

I am not a mathematics student but i have fascination with advance algebra so i read random articles about it without actually understanding it (lol). I'm not sure but I think I read an article a long ...
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### Question about proof of euler's criterion

When a is quadratic residue of the odd prime p, we arrived to the conclusion $(p-1)! \equiv -a^{{p-1}/2}\pmod{p}$. How does that imply $a^{{p-1}/2} \equiv 1\pmod{p}$
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### When can $\frac{3}{n}$ not be written as the sum of two reciprocals of natural numbers?

Show that the set of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S = \left\{n \mid \frac{3}{n} \neq \frac{1}{p}+\frac{1}{q}\right\}$...
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### Proving Wilson's theorem

Wilson's theorem: if $p$ is prime then $(p-1)! \equiv -1(mod$ $p)$ Approach: $$(p-1)!=1*2*3*....*p-1$$ My teacher said in class that the gcd of every integer less than p and p is 1, so every ...
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### Prove this Inequality using Induction

Prove using induction that $|\prod_{j=1}^{n}a_{j} - 1| \leq \sum_{j=1}^{n}|a_{j}-1|$ for $|a_{j}| \leq 1$. So far, I have the Base Case: When $n=1$, we have $|a_{1} - 1| = |a_{1} - 1|$. The ...
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### Reducing an indicator function summation into a simpler form.

Context I am attempting to reduce the space I need to store in an array in a program. I have made it so that the indices are always sorted. There are no indices where they are equal, and no indices ...
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### $\frac{\phi(m)}{m}$ is dense in $[0,1]$

Let $n$ be a natural number, $n \geq 2$, and let $\phi$ be Euler's function; i.e. $\phi(n)$ is the number of positive integers not exceeding $n$ and coprime to $n$. Given any two real numbers $\alpha$ ...
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### What is a Diophantine equation, and why should we care about them? [closed]

As the question title suggests, what is a Diophantine equation, and why should a high schooler learning about elementary number theory care about them?
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### Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers [closed]

As so far as usage in elementary number theory goes, what is the difference between the natural numbers, the integers, the rational numbers, the complex numbers, and the Gaussian integers?
Can Miller's Test be replaced with the bound below in hopes that it would make a faster general-purpose primality test (compared to ECPP). If $n$ is an $a$-SPRP for all primes $a$ $<$ ($\log_2 n$)...