Large runs of Mersenne composites

[manasi]

Hi! I am new to this forum and still clueless about how to use manual testing. I tried to post some observations earlier in manual testing. I am not sure whether this current post is necessary.

For any given n ∈Ν, there exists infinitely many runs of Mersenne composites.

Let {2=p_1,3=p_2,…,p_k} be all the listed primes ≤n.

Then { 2^(n!+j)-1∶ 2≤j≤n } ,

{ 2^(2^l1 3^l2….n^ln+j)-1∶2≤j≤n ,l_1,l_2,…,l_n∈Ν},

{ 2^(2^l1 3^l2….〖pk〗^lk (p_2-1)^n2 (p_3-1)^n3…(p_k-1)^nk+j)-1∶ 2≤j≤n+1 ,l_1,l_2,…,l_n∈Ν, n_2,…,n_k∈Ν }

[manasi]

I wanted to edit the above post

l_i = max { t_i : [n/ t_i ] }

[science_man_88]

Originally Posted by manasi

Hi! I am new to this forum and still clueless about how to use manual testing. I tried to post some observations earlier in manual testing. I am not sure whether this current post is necessary.

For any given n ∈Ν, there exists infinitely many runs of Mersenne composites.

Let {2=p_1,3=p_2,…,p_k} be all the listed primes ≤n.

Then { 2^(n!+j)-1∶ 2≤j≤n } ,

{ 2^(2^l1 3^l2….n^ln+j)-1∶2≤j≤n ,l_1,l_2,…,l_n∈Ν},

{ 2^(2^l1 3^l2….〖pk〗^lk (p_2-1)^n2 (p_3-1)^n3…(p_k-1)^nk+j)-1∶ 2≤j≤n+1 ,l_1,l_2,…,l_n∈Ν, n_2,…,n_k∈Ν }

you could just use LaTeX and make this all look nice.

[Batalov]

Originally Posted by manasi

For any given n ∈Ν, there exists infinitely many runs of Mersenne composites.

Mersenne numbers can only prime for prime indices, so what you are saying is obvious because there are infinitely many runs of composites of length > n.

Read first, post later.
For example, you can start by reading this.