Description: The fifth Mersenne number M_5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | m5prm | ⊢ ( ( 2 ↑ 5 ) − 1 ) ∈ ℙ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3nn0 | ⊢ 3 ∈ ℕ0 | |
2 | 2nn0 | ⊢ 2 ∈ ℕ0 | |
3 | 1nn0 | ⊢ 1 ∈ ℕ0 | |
4 | 2exp5 | ⊢ ( 2 ↑ 5 ) = ; 3 2 | |
5 | 3p1e4 | ⊢ ( 3 + 1 ) = 4 | |
6 | 2m1e1 | ⊢ ( 2 − 1 ) = 1 | |
7 | 1 2 3 4 5 6 | decsubi | ⊢ ( ( 2 ↑ 5 ) − 1 ) = ; 3 1 |
8 | 31prm | ⊢ ; 3 1 ∈ ℙ | |
9 | 7 8 | eqeltri | ⊢ ( ( 2 ↑ 5 ) − 1 ) ∈ ℙ |