who can prove that 2^n −1produces a prime number when n is a prime number


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Question Number 165322 by Ari last updated on 29/Jan/22

$w h o c a n p r o v e t h a t 2^{n} - 1 p r o d u c e s$ $a p r i m e n u m b e r w h e n n i s a$ $p r i m e n u m b e r$
Commented by mr W last updated on 30/Jan/22

$n \neq p r i m e \Rightarrow 2^{n} - 1 \neq p r i m e$ $b u t n = p r i m e ⇏ 2^{n} - 1 = p r i m e$
Commented by mr W last updated on 30/Jan/22

$p r i m e s o f t y p e 2^{n} - 1 w i t h n = p r i m e$ $a r e c a l l e d M e r s e n e p r i m e s . i t {d o e s n}^{'} t$ $m e a n t h a t 2^{n} - 1 i s a l w a y s p r i m e f o r$ $a n y p r i m e n . 2^{n} - 1 i s p r i m e o n l y f o r$ $s o m e p r i m e s n . t h e l a r g e s t k n o w n n$ $i s 82 589 933 w h i c h w a s f o u n d i n 2018 .$
	Answered by JDamian last updated on 29/Jan/22

	$n o b o d y$ $2^{11} - 1 = 2047 = 23 \times 89$
	Commented by mr W last updated on 30/Jan/22

	$m o r e c o u n t e r e x a m p l e s :$ $2^{23} - 1 = 8388607 = 47 \times 178481$ $2^{29} - 1 = 536870911 = 233 \times 1103 \times 2089$ $2^{37} - 1 = 137438953471 = 223 \times 616318177$
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