The Mersenne number Mn is composite whenever n is composite

The Mersenne number Mn is composite whenever n is composite. For the composite number n= 10, find Mn-2n-1.

Solution

1) For n = 10 , Mersenne number M_n = 2ⁿ - 1

    M₁₀ = 2¹⁰ - 1

   M₁₀ = 1024 - 1

   M₁₀ = 1023

2 )   

For n = 9 , Mersenne number M_n = 2ⁿ - 1

M₉ = 2⁹ - 1

M₉ = 512 - 1

M₉ = 511

3 )

a ) \" Sixth \" fermat number is ,2^{2^5} + 1 = 2 ³² + 1

= 4294967296 + 1

= 4294967297

Therefore,

   2^{2^5} + 1 = 4294967297

b )   ( 2^{2^5} + 1 ) / 641 =  4294967297 / 641

= 6700417

4 )

a ) Fermat number F₄ : 2^{2^n} + 1 for n = 4

F₄ = 2^{2^4} + 1

= 65536 + 1

= 65537

Therefore,

F₄ = 65537