Number Theory

Whereas uncompressed public keys have a prefix of 04 ,compressed public keys start with either a 02 or a 03 prefix. Let’s look at why there are two possible prefixes: because the left side of the equa...

A quadratic residue (mod p) refers to an integer c such that there exists an integer x for which the congruence x^2 \equiv c (mod  p) holds, right?Rewriting:x^2(mod  p) \equiv c (mod  p)Which means t...

Suppose for $k\in\Z$, $m,n\in\Z^+$, $\gcd(k,n)=\gcd(k,m)=\gcd(m,n)=1$, where $k$ has the orders of 5 modulo m, and 7 modulo n$. Find the order of k modulo mn and justify your solution.

Solve the congruence $x^{83}\equiv 7\pmod {139}$. Please explain your working at each stage as if you were teaching an undergraduate student who were learning this for the first time! I am really tryi...

Thanks a lot!

(See the paper in the attatchment for a background behind the problems.)Suppose $A=\mathbb{Q}$ where $f:A\to\mathbb{R}$ such that:$f(x)= \begin{cases} 1 & x\in\left\{{(2n+1)}/{2m}:n\in\mathbb{Z},...

I am familiar with the method of exponentiation by squaring. However, since I am only interested in the last 6 digits, it seems unnecessary to calculate the whole number.Is there a method to do this e...

Prove that $p^2-1$ is divisible by 24 for any prime number $p > 3$. Can this be proved with an elementary argument? 

Bases and Number Representations question, working out please.One byte is an 8-bit number; for example, 00000001. A byte can represent decimal numbers from 0 to 255, and is historically the smallest a...

Find the solutios or prove that there is no solutionA) x^2 ≡ 13 (mod 17)B) x 2 ≡ 14 (mod 17)

 Question I want to find a measure of uniformity that determines the most "uniform" "structure" for the most cases of countably infinite sets?  Definition of Structure A "structure" of a countably inf...

If both $n$ and $\sqrt{n^2+204n}$ are positive integers, find the maximum value of $n$. 

Let $S$ be a subset of $\{1,2, \dots, 2n\}$ with $(n+1)$ elements. Prove that there is an element of $S$ that is divisible by another element of $S$.

I am trying to find the integers $a,b$ and $c$ so that the number of hidden cubes in a rectangular cube with dimensions $a,b,c$ (built from $1\times 1$ cubes) to be half of the total number of cubes....