It is easy to show that every natural number $n$ can be written as $n = \frac{xy}{x+y}$ with $x,y\in \mathbb{N}$ by setting $x = y = 2n$. Now I experimented a little bit with numbers of the form $n=\frac{xyz}{x+y+z}$ and it seems that every natural number $n$ can be written this way with $x,y,z \in […]

I know that I need to reduce $7^{12341} \pmod {1000}$ By Euler I have $7^{\phi(1000)}\equiv 7^{400}\equiv1\pmod{1000}$ That leaves me with the monster $7^{341}\pmod{1000}$ Is there a way to reduce this smoothly without working $7^2, 7^4, 7^8$ etc manually ?

I’m working on some Chinese Remainder problems and it doesn’t seem like I have the procedure down correctly. I’ll list the steps I’m taking so hopefully someone can spot what it is I’m doing wrong. Find the least nonnegative solution of each system of congruences below. $x \equiv 3 \space mod \space 4$ $x \equiv […]

I made a proof by contradiction. Suppose $δ=(a+b,\operatorname{lcm}[a,b])$ and let it be that $δ\neq(a,b)$. Then $\exists ε\big(ε=(a,b) \land ε\gt δ \big) \implies ε|a \land ε|b \implies ε|(a+b)$. It is also true that $ε|\operatorname{lcm}[a,b]$. By the two previous statements, we get that $ε|(a+b,\operatorname{lcm}[a,b])\implies ε|δ$. This is absurd since $ε>δ$. Thus $δ=(a,b)$. Is it correct? I wonder […]

This problem is taken from Ivan Niven’s “An Introduction to the Theory Of Numbers”. Show that ${{p^\alpha-1}\choose{k}} \equiv ({-1})^k\pmod p$. Note: This is not similar to this one, as $k! | p^\alpha$ is possible. My Attempt: We proceed by induction on k: Let, ${{p^\alpha-1}\choose{k-1}}=r$.Let $k=p^t*q \ : t<\alpha$. $(k,p^\alpha-k)=p^t$. $${{p^\alpha-1}\choose{k}} = {{p^\alpha-1}\choose{k-1}}*\frac{p^\alpha-k}{k}={{p^\alpha-1}\choose{k-1}}*\frac{p^{\alpha-t}-q}{q}$$. So, by induction […]

Find $v_p\left(\binom{ap}{bp}-\binom{a}{b}\right)$, where $p>a>b>1$ and $p$ odd prime. Here $v_p(k)$ denotes the largest $\alpha\in\mathbb Z_{\ge 0}$ s.t. $p^\alpha\mid k$. We have $p\nmid\binom{ap}{bp}$ and $p\nmid \binom{a}{b}$. $$\binom{ap}{bp}-\binom{a}{b}=\frac{(ap)!b!(a-b)!-a!(bp)!(ap-bp)!}{(bp)!(ap-bp)!b!(a-b)!}$$

Let $c$ be a $3^n$ digit number whose digits are all equal. Show that $3^n$ divides $c$. I have no idea how to solve these types of problems. Can anybody help me please?

If a number can be expressed as a product of n unique primes, in how many ways can the number be expressed as a difference of two squares?

Given e and d as the encryption and decryption component respectively, textbook RSA has the property $ed\equiv 1\pmod {\phi(n)} $. The requirement is that suppose there is another function $\lambda(n)$ such that $$\lambda(n) = {\phi(n)\over gcd (p-1, q-1)}$$ and $$ed\equiv 1\pmod{\lambda(n)}$$I need to prove that e and d still work as encryption and decryption components. […]

If $r$ is a primitive root of odd prime $p$, prove that $\text{ind}_r (-1) = \frac{p-1}{2}$ I know $r^{p-1}\equiv 1 \pmod {p} \implies r^{(p-1)/2}\equiv -1 \pmod{p}$ But some how I feel the question wants me prove the result using indices properties (w/o factoring.. ) So I am posting it here to see if there are […]

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