help@askiitians.com

+91-87964 74404

Grade 12Algebra

if 2^n+1 is prime then prove that n isa power of 2

ashwen venkatesh

12 Years agoGrade 12

1 Answer

Sher Mohammad

Approved Tutor Answer12 Years ago

$a^n-b^n=(a-b)\sum_{k=0}^{n-1} a^kb^{n-1-k}.$

If $n$ is a positive integer but not a power of 2, then $n = rs$ where $1 \le r < n$ , $1 < s \le n$ and $s$ is odd.

By the preceding lemma, for positive integer $m$ ,

$(a-b) \mid (a^m-b^m)$

where $\mid$ means "evenly divides". Substituting $a = 2^r$ , $b = -1$ , and $m = s$ and using that $s$ is odd,

$(2^r+1) \mid (2^{rs}+1),$

and thus

$(2^r+1) \mid (2^n+1).$

Because $1 < 2^r+1 < 2^n+1$ , it follows that $2^n+1$ is not prime. Therefore, by contraption $n$ must be a power of 2.

sher mohammad

b.tech, iit delhi

What is a binomial expression?

algebra

1 Answer Available

Last Activity: 2 Years ago

Can we divide two vectors? If this is possible then how?

algebra

1 Answer Available

Last Activity: 2 Years ago

How do you write “the product of 18 and q” as an algebraic expression?

algebra

1 Answer Available

Last Activity: 2 Years ago

दहावीच्या आय सी एस इ विद्यार्थ्यांनी पुढे काय करायला पाहिजे

algebra

1 Answer Available

Last Activity: 3 Years ago

Find the least number of square tiles by which the floor of a room of dimensions 16.58 m and 8.32 m can be covered completely. why HCF and not LCM as “least” keywor dis given?

algebra

1 Answer Available

Last Activity: 3 Years ago

View all Questions