Conjecture: Given any fixed $n \geq 0, k \geq 1$, there always exists a solution $q_i, i = 1..n, \ p_j, p_k, j \gt k$ in the odd primes to: $$ 2^k q_1 \cdots q_n = p_j - p_k$$ Notice, I don't ask for infinitude of solutions, or for any such $2^k q_1\cdots q_n$ to appear as a prime gap. Can we...

Is the following conjecture true? If $a^n\equiv{1}\mod{p}$, where $p$ is prime, and $a$ is not divisible by $p$, then the least value of $n$ is a factor of $p-1$. Fermat's Little Theorem tells us that $n=p-1$ is always a solution, but it doesn't tell us when there are smaller solutions. For exa...

Before beginning in explaining my pathetic attempt to solve Collatz Conjecture, I am a High-School student, so this is built based on intuition. Some backstory I was playing with Collatz Conjecture for a while now. I discovered that any number 'x' where $$x=2^n,n\epsilon Z^+$$ is "collapsed" to t...