Intereting Posts

Unique manifold structure
Find the linear fractional transformation that maps the circles |z-1/4| = 1/4 and |z|=1 onto two concentric circles centered at w=0?
The sections of the projection $\bigsqcup_{i:I} X_i \rightarrow I.$
$n$ positive integer, then $n=\sum_{d|n} \phi(d)$ (proof Rotman's textbook)
A simple permutation question – discrete math
Legendre symbol $(-21/p)$
How many absolute values are there?
Does the correctness of Riemann's Hypothesis imply a better bound on $\sum \limits_{p<x}p^{-s}$?
Implicit Differentiation Help
Why does the Dedekind Cut work well enough to define the Reals?
$4$ or more type $2$ implies $3$ or less type $1$
The primitive spectrum of a unital ring
A generalization of Kirkman's schoolgirl problem
When does the topological boundary of an embedded manifold equal its manifold boundary?
Image of the union and intersection of sets.

I’m in the process of proving Fermat’s little theorem.

For a prime integers $p$ we have $a^p \equiv a \mod{p}$

I proved it for a non-negative $a$, not I need to generalize the case to an arbitrary $a \in \mathbb{Z}$. That is, I need to prove that give a negative integer $a$ we have $a^p \equiv a \mod{p}$ using the fact that it is so for a non-negative $a$.

- solve $3x^2 + 6x +1 \equiv 0 \pmod {19}$
- If $n\equiv 4 \pmod 9$ then $n$ cannot be written as sum of three cubes?
- Using Fermat's little theorem to find remainders.
- Flaw or no flaw in MS Excel's RNG?
- I finally understand simple congruences. Now how to solve a quadratic congruence?
- Number Theory: Solutions of $ax^2+by^2\equiv1 \pmod p$

- Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
- Proving that an integer is even if and only if it is not odd
- Solution to congruence $z^2=c$ mod n
- Prove or disprove that $\phi(a^n - 1)$ is divisible by n
- Proof: How many digits does a number have? $\lfloor \log_{10} n \rfloor +1$
- Can we prove this inequality in another way?
- Is gcd the right adjoint of something?
- Is $n^{th}$ root of $2$ an irrational number?
- Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
- IMO 1987 - function such that $f(f(n))=n+1987$

Use the fact that if $a^p\equiv a$ for some $a$, then this automatically also holds for every $a’$ that is congruent to $a$. From first principles, here is how it goes:

If $a$ is negative, then there is still a $k$ such that $a+kp$ is positive. Then we have

$$ (a+kp)^p \equiv a+kp \pmod p $$

The right-hand side obviously equals $a$ modulo $p$. For the left-hand side, expand using the binomial theorem — all terms except for $a^p$ include one or more factors of $p$ and are therefore $0$ modulo $p$. So the left-hand side is congruent to $a^p$ modulo $p$.

- Non-invertible elements form an ideal
- Dual space of $H^1$
- An Integral involving $e^{ax} +1$ and $e^{bx} + 1$
- Integral $I=\int \frac{dx}{(x^2+1)\sqrt{x^2-4}} $
- Euler's Approximation of pi.
- Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input
- Calculus of Natural Deduction That Works for Empty Structures
- Limit of sequence in which each term is defined by the average of preceding two terms
- Question on showing a bijection between $\pi_1(X,x_0)$ and $$ when X is path connected.
- Quotient topology by identifying the boundary of a circle as one point
- Riemann Hypothesis and the prime counting function
- clock related challenge
- Find an integer such that when squared, the first 4 digits are '6666'.
- How to evaluate these indefinite integrals with $\sqrt{1+x^4}$?
- Indicator function for a vertex-induced random subgraph of $G$?