Mersenne

Number Theory Level 1

Read the following statements carefully:

[ 1 ] [1] . For all prime numbers p p , 2 p − 1 2^p-1 is a prime number.

[ 2 ] [2] . If p p is a composite number, then it is impossible for 2 p − 1 2^p-1 to be a prime number.

[ 3 ] [3] . Statement number [ 1 ] [1] isn't true.

Which of these statements is/are correct?

[ 1 ] [1] [ 1 ] [1] and [ 2 ] [2] [ 2 ] [2] and [ 3 ] [3] None of the above

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Mursalin Habib by Mursalin Habib , 23, Bangladesh

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2 solutions

Etienne Duclos
Apr 10, 2014

2 11 − 1 = 2047 = 23 ∗ 89 2^{11}-1 = 2047 = 23*89 is not prime, so [ 1 ] [1] is false and [ 3 ] [3] is true.

If p p is a composite number, let p = a b p = ab . Then 2 p − 1 = 2 a b − 1 2^p -1 = 2^{ab} - 1 = ( 2 a ) b − 1 = ( 2 a − 1 ) ∗ ( 2 a ( b − 1 ) + 2 a ( b − 2 ) + 2 a ( b − 3 ) + . . . + 2 a + 1 ) = (2^a)^b - 1 = (2^a - 1)*(2^{a(b-1)} + 2^{a(b-2)} + 2^{a(b-3)} + ... + 2^a + 1) , which is a composite number. So [ 2 ] [2] is true.

Sorry, I will learn LaTeX one day.

[LaTeX edits - moderator]

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Sorry, I will learn LaTeX one day.

You are already half way there. Just put your math inside these brackets \ ( . . . \ ). Don't put any space between them. Try it out!

Mursalin Habib - 7 years, 2 months ago

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latex is good

math man - 6 years, 8 months ago
Curtis Clement
Apr 24, 2015

For [1]: 2 p − 1 i s p r i m e ⟹ p i s p r i m e 2^p -1 \ is \ prime \implies \ p \ is \ prime but the arrow goes only one way so [1] is false and [3] is true. Etienne has already demonstrated that [2] is indeed true.

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