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Question Number 56660 by tanmay.chaudhury50@gmail.com last updated on 20/Mar/19

Commented by kaivan.ahmadi last updated on 20/Mar/19

$c a n y o u p r o o f i t s i r ?$
Commented by tanmay.chaudhury50@gmail.com last updated on 21/Mar/19

$i t h i n k t o p r o v e$ $2^{p r i m e n u m b e r} - 1 = p r i m e n u m b e r$ $w e n e e d f (n) w h i c h i s p r i m e$ $2^{f (n)} - 1 = p r i m e n u m b e r$ $t o m y k n o w l d g e n o s u c h g e n e r a l f o r m u l a f (n)$ $d e r i v e d t i l l n o w w h i c h d e s c r i b e p r i m e n u m b e r .$
Commented by mr W last updated on 21/Mar/19

$n e x t o n e :$ $23 i s p r i m e$ $2^{23} - 1 = 8388207 = 47 \times 178481 \Rightarrow n o p r i m e$ $n e x t o n e :$ $29 i s p r i m e$ $2^{29} - 1 = 536870911 = 233 \times 1103 \times 2089 \Rightarrow n o p r i m e$ $t h e r e f o r e$ $2^{p r i m e} - 1 \underset{b u t n o t a l w a y s}{\overset{m a y b e}{=}} p r i m e$
Commented by mr W last updated on 21/Mar/19

$i f 2^{p r i m e} - 1 = p r i m e w e r e t r u e, t h e n$ $2^{2^{p r i m e} - 1} - 1 = p r i m e i s a l s o t r u e, i . e .$ $i f w e k n o w a p r i m e s a y p, w e w i l l k n o w$ $i n f i n i t e o t h e r p r i m e s :$ $2^{p} - 1, 2^{2^{p} - 1} - 1, 2^{2^{2^{p} - 1} - 1} - 1, . . . .$ $b u t t h i s i s n o t t h e c a s e . i n f a c t t i l l n o w$ $t h e l a r g e s t p r i m e n u m b e r w e k n o w$ $i s 2^{282589933} - 1 . i f y o u c a n f i n d t h e n e x t$ $o n e, y o u r n a m e w i l l g o i n t o t h e h i s t o r y .$
Commented by tanmay.chaudhury50@gmail.com last updated on 21/Mar/19

$b a h d a r u n y h a n k y o u s i r . . .$
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