# Tagged Questions

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### Pre College Mathematics

During my school days I was a very keen student of mathematics. But circumstances led me to opt for commerce at the college level. Now I wish to continue learning mathematics on a self study basis. ...
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### Fermat's last theorem fails in $\mathbb{Z}_p$ for $p$ sufficiently large

Statement For any $n, \;x^n+y^n=z^n$ has non-trivial solutions in $\mathbb{Z}_p$ for all but finitely many $p$. I remember seeing this problem on an first year undergraduate problem ...
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### Is there a commonly studied number structure with “two” of a given number?

I'm wondering whether there are structures of numbers where there is intuitively "two" of a given number. I have in mind something like what is illustrated in the following example number line ...
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### The sum of a square and twice a square

Consider the set $\{3,6,9,11,12,17,18,19,22\ldots\}$ (OEIS A154777) of positive integers that are expressible in the form $a^2 + 2b^2$. Is there a theorem about the form of such numbers analogous to ...
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### Modified Euler's Totient function for counting constellations in reduced residue systems

I am working on a modified totient function for counting constellations in reduced residue systems for the same range that Euler's totient function is defined over. This post is separated into three ...
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### Are there (known) bounds to the following arithmetic / number-theoretic expression?

I apologize in advance if this is something that is already well-known in the literature, but I would like to ask nonetheless (for the benefit of those who likewise do not know): Are there (known) ...
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### If $A\equiv 1\pmod{3}$, then $4p=A^2+27B^2$ uniquely determines $A$.

If $p\equiv 1\pmod{3}$, it's well know that $p$ can be expressed as $$p=\frac{1}{4}(A^2+27B^2).$$ In this letter by Von Neumann, he mentions that Kummer determined that $A$ is in fact uniquely ...
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### Good introductory readings to topics related to prime numbers for non-mathematicians

I'm a maths hobbyist who is fascinated by prime numbers. My quest to delve into the interesting parts of the topic is always hindered by my inability to understand the notation and concepts I ...
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### Need a recommendation for a book that covers congruence classes.

I want to learn about congruence classes and need a book recommendation. With a lot of examples.
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### Number Theory Practice Questions

Can anyone please suggest a book or a link to website where I can find Practice questions (and there solutions) on Number Theory. More specifically questions on the following topics: Divisibility ...
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### Reference request: Clean proof of Fermat's last theorem for $n=3$.

I have seen a proof for FLT, $n=3$ using factorisation in the ring of Eisenstein integers, but it's quite long and convoluted; I am wondering if there is a more 'advanced' proof which avoids infinite ...
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### Parameters giving maximal-length Collatz-like sequence

In a recent question the following recursive sequence was considered: $$a_{n+1} = \cases{\frac{a_n}{2} & a_n is even \\ a_n +d & a_n is odd}, \quad a_1 = d + 1$$ where $d$ is an odd ...
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### Solving simple congruences by hand

When I am faced with a simple linear congruence such as $$9x \equiv 7 \pmod{13}$$ and I am working without any calculating aid handy, I tend to do something like the following: "Notice" that adding ...
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### Multiplication tables with all entries distinct

Let positive integers $\alpha$ and $\beta$ be given. It is easy to find sets $A$ and $B$ of positive integers such that: $|A|=\alpha$ and $|B|=\beta$ The set $P = \{ab : a\in A, b\in B\}$ contains ...
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### Summing over a cyclic subgroup of a multiplicative group mod n

Let $x$ be a unit in $\mathbb Z/ n \mathbb Z$ of multiplicative order $m$. I am trying to determine when it is that $$\sum_{i=0}^{m-1} x^i \equiv 0 \mod n .$$ Is this kind of situation something ...
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### Diophantine equation $x^y-y^x=11$

How can one find all integer solutions to $x^y-y^x=k$, for a given k? Example case $x^y-y^x=11$
A while back I asked a question concerning the existence of squares of the form $1234567891011\ldots$ This question is similar but more general. Let $n,p,\ell$ be positive integers, then we define ...
### When does a prime $q$ not divide $\lambda p +1$ for all $\lambda$?
What can be said about the following question? For which prime numbers $p$ there exists another prime number $q(p)$ such that $q(p)$ does not divide $\lambda p +1$ for all integers $\lambda$ ? For ...