Articles of elementary number theory

Number of solutions of $a^{3}+2^{a+1}=a^4$.

Find the number of solutions of the following equation $$a^{3}+2^{a+1}=a^4,\ \ 1\leq a\leq 99,\ \ a\in\mathbb{N}$$. I tried , $$a^{3}+2^{a+1}=a^4\\ 2^{a+1}=a^4-a^{3}\\ 2^{a+1}=a^{3}(a-1)\\ (a+1)\log 2=3\log a+\log (a-1)\\ $$ This is from chapter quadratic equations. I look for a short and simple way. I have studied maths up to $12$th grade.

Generating all coprime pairs

The Wikipedia article on coprime integers has a brief section on generating all coprime pairs. All pairs of positive coprime numbers $(m,n)$ (with $m>n$) can be arranged in two disjoint complete ternary trees, one tree starting from $(2,1)$ (for even-odd and odd-even pairs), and the other tree starting from $(3,1)$ (for odd-odd pairs). The children […]

How much can a fraction reduce?

Assume $x/a$ and $y/b$ are positive fractions in it’s reduced form. If $x/a+y/b=z/c$, where $z/c$ is also reduced. What can we say about $c$? Does $\frac{ab}{\gcd(a,b)^2}|c$? If it’s not true. Is it true when they are all between 0 and 1?

Find all values of $p$ such that $ax^2+bx+c \equiv 0 (\bmod p)$ have solution

Is there a general way to find all values of $p$ such that the congruence $ax^2+bx+c \equiv 0 (\bmod p)$ have solution, we can assume that $ax^2+bx+c =0 $ have solution.

Primes $p$ such that $3$ is a primitive root modulo $p$ , where $p=16 \cdot n^4+1$?

How to prove following statement : Conjecture : Let $p$ be a prime number of the form : ${\color{BlueViolet}{p=16 \cdot n^4+1}}$ If $n$ is an odd prime greater than $3$ then $3$ is a primitive root modulo $p$ . I wrote small Maple program (see below) in order to find counterexample , but I haven’t […]

Basic number theory proofs

Deduce that there is a prime gap of length $\geq n$ for all $n \in \mathbb{N}$ Show that if $2^n – 1$ is prime, then $n$ is prime. Show that if $n$ is prime, then $2^n – 1$ is not divisible by $7$ for any $n > 3$. I’m not really sure how to do […]

Prove $a^{pq} \equiv a \pmod {pq}$

For p and q being distinct primes. I am given that $a^{p} \equiv a \pmod q$ and $a^q \equiv a \pmod p$. We are supposed to use the Fermat’s Theorem and Chinese remainder Theorem to Prove. I have assumed that $p$ not equal to $q$ is what is meant by distinct. I know $a^p \equiv […]

Smallest $k$ s.t. $7x+1=9y+2=11z+3=k$, all positive integers

Find the smallest positive integer, which on dividing with 7 gives remainder 1, on dividing with 9 gives (remainder) 2 and that after division by 11 yields 3 as remainder. i.e., find smallest $k \in \mathbb{Z}^+$ such that $$k=7x+1=9y+2=11z+3 \ x,y,z \in \mathbb{Z}^+$$. The answer is 344 which I got by observing each number. First […]

Finding $26$th digit of a number.

I have found an old Regional Olympiad problem (It is so old that you can call it ancient). Which is as follow: $N$ is a $50$ digit number (in the decimal scale). All digit except the $26$th digit (From the left) are $1$. If N is divisible by $13$ then find the $26$th digit. My […]

Are all known $k$-multiperfect numbers (for $k > 2$) *not* squarefree?

A positive integer $N$ is said to be $k$-multiperfect if $$\sigma(N) = kN$$ where $\sigma(x)$ is the sum of the divisors of $x$ and $k$ is a positive integer. (The case $k = 2$ reduces to the original notion of perfect numbers.) Now my question is the following: Are all known $k$-multiperfect numbers (for $k […]