Intereting Posts

Let V denote the Klein 4-group. Show that $\text{Aut} (V)$ is isomorphic to $S_3$
Finite Cartesian Product of Countable sets is countable?
Riemannian metric of the tangent bundle
Find remaining vertices of a square, given 2
Why is it that Complex Numbers are algebraically closed?
A question on behavior of a function which is a limit of a sequence of functions converging under some norm
Solving a recurrence by using characteristic equation method
What is combinatorics?
Is the “binary operation” in the definition of a group always deterministic?
Proving that a linear isometry on $\mathbb{R}^{n}$ is an orthogonal matrix
fair die or not, from 3D printer
Integral domain, UFD and PID related problem
Kronecker product and outer product confusion
If $(a_n)$ is a decreasing sequence of strictly positive numbers and if $\sum{a_n}$ is convergent, show that $\lim{na_n}=0$
Remainders of polynomials of higher degrees.

This came up as a part of algorithm puzzles:

Given a number $N$, how to find the prime $P$ such that $P<N$ and the difference $N-P$ is minimum.

For small $N$, simple sieves do work, but I’m unable to find solution for large values of $N$.

I tried to modify the sieves to somehow look over large values only, didn’t get anywhere.

Any solutions/hints for this?

- Intervals that are free of primes
- Counting non-negative integral solutions
- Generalised Hardy-Ramanujan Numbers
- Show that there are infinitely many positive integers $N$ that cannot be written in the form $a^n+b^n+c^n$
- Are those two numbers transcendental?
- Order of cyclic groups and the Euler phi function
- Prime numbers are related by $q=2p+1$
- A subset whose sum of elements is divisible by $n$
- last $2$ digit and last $3$ digit in $\displaystyle 2011^{{2012}^{2013}}$
- Status of a conjecture about powers of 2

Start counting downward from $N-1$ After trial division by some small primes, you can use one of a number of probabilistic tests like the Fermat primality test. If $p$ is prime, $a^{p-1} \equiv 1 \pmod p$. This can be checked quickly. If it is true for a few $a$, the chance of $p$ being composite is very small. If you insist on certainty, you can finish up with one of the deterministic tests.

- Show that orthogonal matrices have eigenvalues with magnitude $1$ without a sesquilinear inner product
- Convergence in product topology
- Expectation of pairwise differences between uniform random points in hypercube
- Proof that $\dim(U_1+U_2) = \dim U_1 + \dim U_2 – \dim(U_1\cap U_2)$
- Integrating around a dog bone contour
- is division by zero automatically irrational?
- Category Theory usage in Algebraic Topology
- Maximizing the trace
- Contradiction: Prove 2+2 = 5
- Why is the infinite set from the axiom of infinity the natural numbers?
- Klein-bottle and Möbius-strip together with a homeomorphism
- Beautiful cyclic inequality
- what are the possible answers we can get for the below intergral?
- Why does $0.75 \times 1.25$ equal $0.93$ and not $1$?
- Trying to understand the use of the “word” pullback/pushforward.