0
votes
1answer
77 views

Have you seen this theorem before? (GCD divides, neccessary & sufficient condition)

Conjecture. Let $a,b, c\in \Bbb{Z}, b \neq 0$, The following conditions are equivalent: (1) $d = \gcd(a,b)$ divides c. (2) There's a polynomial in $f \in \Bbb{Z}[X,Y]$ with $c$ constant term, such ...
-1
votes
1answer
61 views

Efficient way to find lowest divisor of an integer.

I have followed the given way to find the lowest divisor of an integer, Let us assume n is the given integer. Check n is ...
6
votes
1answer
70 views

Is this divisibility test for 4 well-known?

It has just occurred to me that there is a very simple test to check if an integer is divisible by 4: take twice its tens place and add it to its ones place. If that number is divisible by 4, so is ...
5
votes
1answer
109 views

Does this have a name: If an odd prime $p$ does not divide $a$, then $p$ divides $a^n + 1$ or $a^n - 1$

After seeing and doing a bunch of proofs like "For all $a$ in the natural numbers, then if $7$ does not divide $a$, then $7$ divides $a^3+1$ or $a^3-1$," I conjectured the following, but got stuck in ...
1
vote
1answer
71 views

Looking for references on results on powers of primes dividing $y^n-1$

For a prime $p$ and positive integer $n$, let $E(n,p)$ be the greatest $k$ such that $p^k \mid n$, and $E(n,p) = 0$ if $p \nmid n$. Let $E(n) = E(n, 2)$. A number of years back, I proved the ...
3
votes
1answer
158 views

Good resource of maths problems with solutions

I'm searching for a good book or web page that has a good amount of problems and their solutions, at undergraduate level, of divisibility, inequalities, induction, etc. Thanks in advance