We can't be prime together

\[\nexists~a,b\in\Bbb Z^+~(a\neq b)~\mid~a+x\in\Bbb P\iff b+x\in\Bbb P~\forall~x\in\Bbb Z_{\geq 0}\]

Prove the statement above.

Details and Assumptions :

$\Bbb P$ denotes the set of all prime numbers.
$\Bbb Z^+$ denotes the set of all positive integers.
$\Bbb Z_{\geq 0}=\Bbb Z^+\cup \{0\}$ denotes the set of all non-negative integers.

Source : Math StackExchange

#NumberTheory #PrimeNumbers

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Comments

Seems simple if you use contradiction.

Suppose there does exist such an $a$ and $b$ . WLOG, let $a > b$ , and $a - b = d$ . Now let $p$ be the smallest prime such that $p > a$ .

Then, by induction,we can prove all $p + nd$ for all natural numbers $n$ .

This implies that in every set of $d$ consecutive integers, starting from an integer larger than $p$ , we can find at least one prime.

But this is not true for the set $k(d+1)! + 2, k(d+1)! + 3, ... k(d+1)! + (d+1)$ , in which all elements are composite, and the first element is larger than $p$ by taking some suitable k. Hence, we have a contradiction.

Siddhartha Srivastava - 5 years, 6 months ago

Yes, that is the intended proof. +1

However, you can cut the proof short by taking $n=p$ which would result in us getting that $p+pd=p(1+d)$ is prime. Contradiction.

The result follows.

Prasun Biswas - 5 years, 6 months ago

Hey seniors and respected professionals, please do comment in this note. I see that nowadays the scope of Brilliant has been limited to just JEE syllabus and Olympiad Mathematics, please try to promote Higher Mathematics too.

Swapnil Das - 5 years, 6 months ago

This problem might well fit into the Olympiad math style actually. There's a really elegant proof (by contradiction) of this that uses just elementary mathematical induction.

Prasun Biswas - 5 years, 6 months ago

Really? Cool then..

Swapnil Das - 5 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$