Is (2^82589933-243)/19^2 prime?
 

Could you proof that (2^82589933-243)/19^2 is composite?
No factor below 4*10^9

[QUOTE=enzocreti;514579][B]Is (2^82589933-243)/19^2 prime?[/B][/QUOTE]
Bet you $100 against your 1 lira that it isn't.

How did you pull that number out of your ear or rear?

[QUOTE=Batalov;514580]Bet you $100 against your 1 lira that it isn't.[/QUOTE]

I assume your bet is that the number is composite?

[QUOTE=enzocreti;514579]Could you proof that (2^82589933-243)/19^2 is composite?
No factor below 4*10^9[/QUOTE]

The number is 24.8 [I]million[/I] digits long it could have factors that are millions of digits long (up to 12.4 million digits), the fact that it has no factor below 4,000,000,000 barely changed the probably that it is prime at all.
You could trial factor it to 4,000,000,000,000,000,000,000 without any factor, and it would still most likely be composite.

[QUOTE=enzocreti;514579]Could you proof that (2^82589933-243)/19^2 is composite?
No factor below 4*10^9[/QUOTE]

Isn't [$]2^{82589933}-1[/$] a Mersenne Prime? Is it true in general that for each Mersenne Prime [$]M[/$], then [$]\dfrac{M-242}{19^2}[/$] is also prime?

I'm just asking out of curiousity maybe this is a stupid question, but how do we see that this number is in Z?

[QUOTE=dcheuk;514616]I'm just asking out of curiousity maybe this is a stupid question, but how do we see that this number is in Z?[/QUOTE]

2[SUP]82589933[/SUP] mod 361 = 243

[QUOTE=GP2;514617]2[SUP]82589933[/SUP] mod 361 = 243[/QUOTE]

Sorry kinda slow lol.

[$$]
\begin{align*}
2^{82589933}&\equiv2^{\left\lfloor82589933/\phi(19^2)\right\rfloor}\pmod{19^2}\\
&\equiv 2^{\left\lfloor82589933/(19^2-19)\right\rfloor}\pmod{19^2}\\
&\equiv 2^{11}\pmod{19^2}\\
&\equiv 243\pmod{19^2}
\end{align*}[/$$]

[QUOTE=dcheuk;514619][$$]\equiv 243\pmod{19^2}[/$$][/QUOTE]

Most programming languages (other than C and C++) have modular exponentiation library functions.

For instance, in Python: [c]pow(2,82589933,19*19)[/c] gives [c]243[/c]

[QUOTE=dcheuk;514619]Sorry kinda slow lol.

[$$]
\begin{align*}
2^{82589933}&\equiv2^{\left\lfloor82589933/\phi(19^2)\right\rfloor}\pmod{19^2}\\
&\equiv 2^{\left\lfloor82589933/(19^2-19)\right\rfloor}\pmod{19^2}\\
&\equiv 2^{11}\pmod{19^2}\\
&\equiv 243\pmod{19^2}
\end{align*}[/$$][/QUOTE]Not sure but isn't it mod instead of floor division?

[tex]\begin{align*}
2^{82589933}&\equiv2^{82589933\pmod{\phi(19^2)}}\pmod{19^2}\\
&\equiv 2^{82589933\pmod{(19^2-19)}}\pmod{19^2}\\
&\equiv 2^{11}\pmod{19^2}\\
&\equiv 243\pmod{19^2}
\end{align*}[/tex]

[QUOTE=retina;514622]Not sure but isn't it mod instead of floor division?

[tex]\begin{align*}
2^{82589933}&\equiv2^{82589933\pmod{\phi(19^2)}}\pmod{19^2}\\
&\equiv 2^{82589933\pmod{(19^2-19)}}\pmod{19^2}\\
&\equiv 2^{11}\pmod{19^2}\\
&\equiv 243\pmod{19^2}
\end{align*}[/tex][/QUOTE]

Yeah you're right my bad.

To correct mine: [$]2^{82589933}\equiv 2^{82589933-\left\lfloor82589933/(19^2-19)\right\rfloor\cdot(19^2-19)}\pmod{19^2}[/$] should do it. :smile: