Intereting Posts

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If $\;p=m+n$ where $p\in\mathbb P$, then $m,n$ are coprime, of course. But what about the converse?

Conjecture:

$p$ is prime if $\;\forall m,n\in\mathbb Z^+\!:\,p=m+n\implies \gcd(m,n)=1$

- About the hyperplane conjecture.
- Closed form for $\int_{-1}^1\frac{\ln\left(2+x\,\sqrt3\right)}{\sqrt{1-x^2}\,\left(2+x\,\sqrt3\right)^n}dx$
- Conjecture $\int_0^1\frac{dx}{\sqrtx\,\sqrt{1-x}\,\sqrt{1-x\left(\sqrt{6}\sqrt{12+7\sqrt3}-3\sqrt3-6\right)^2}}=\frac\pi9(3+\sqrt2\sqrt{27})$
- Complete this table of general formulas for algebraic numbers $u,v$ and $_2F_1\big(a,b;c;u) =v $?
- Every prime number divide some sum of the first $k$ primes.
- List of generally believed conjectures which cannot all be true

Tested (and verified) for all $p<100000$.

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- Is the 3x+1 problem solved?
- Conjecture about integral $\int_0^1 K\left(\sqrt{\vphantom1x}\right)\,K\left(\sqrt{1-x}\right)\,x^ndx$
- How often does $D(n^2) = m^2$ happen, where $D(x)$ is the deficiency of $x$?
- suppose $\gcd(a,b)= 1$ and $a$ divides $bc$. Show that $a$ must divide $c$.
- $\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$
- making mathematical conjectures

It is true. Suppose $p\geqslant 2$ is *not* prime. Then we can write $p=xy$ with $x,y\geqslant 2$. Then we find $p=m+n$, with $m=x$ and $n=x(y-1)$. Those are obviously not coprime.

If $d \mid p$ and $d<p$, then $1 = \gcd(d, p-d) = \gcd(d, p) = d$, so $p$ is prime.

for p = 1 obviously wrong

(for all positive integers m, n with m+n=p (of course there are no ones, doesn’t matter) there is gcd(m,n)=1, but 1=p is not prime)

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