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Number Theory Level 2

Is ${ 2 }^{ 57885161 }-1$ a prime number, or not?

Maybe Yes No Who told you to ask such a question

5 solutions

Omkar Kamat
Dec 30, 2014

Greatest Internet Mersenne Prime Search, or GIMPS, is a collaborative project in which people try and break the world record for the largest known prime. The number shown is currently the largest prime known as of 2014.

Vishal S
Dec 30, 2014

2^odd number - 1 is always a prime number.Since the given power is odd

therefore the given number is a prime number.

What about ${2}^{9} - 1$ , it is $511$ . Which has divisors - $1, 7, 73, 511$

Vatsalya Tandon - 6 years, 5 months ago

One could argue your sentence in two different ways. 1. IF the exponent k is composite, THEN the number 2^k-1 is composite. Since an odd number isn't necessarily prime, you are wrong. 2 At this point you could say that for k prime, 2^k-1 is prime, too, but also saying this would be an error. We currently don't even know if there are infinite primes in the form of 2^p-1, and the relative hypothesis is known as Mersenne (open) conjecture. So definitely nope. That's not true.

Giacomo Hermes Ferraro - 6 years, 5 months ago

Shivam Ach
Jan 10, 2015

so is there any logical answer for this qeustion other than geussing. for 2^11-1 the prime factors are 23*89.. we can observe 23 and 89 both are prime numbers. if we think that 2^n-1 in such form that n being a prime no.{ which is not applicale for n=11}

Aditya Singh
Jan 2, 2015

whnevr 2 is wid a pwr of of an odd integer nd we subtrct 1 frm it we wll get a prime nmbr..for eg.. 2^3 -1=7 2^5-1=31 2^7-1=127

Read my comment above.

Vatsalya Tandon - 6 years, 5 months ago

Vatsalya Tandon
Dec 30, 2014

It is the largest found Mersenne Prime, that is a prime number in the form of ${2}^{n} - 1$

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