## Number Theory

The following section presupposes the results established in the chapter on Set Theory.

Collatz theorem: The sequence n/2 for even n and 3n + 1 for odd n always ends at 1 for n ∈ ℕ*.

Proof: After one iteration, by (Jeffrey C. Lagarias, "The 3x+1 problem and its generalizations", American Mathematical Monthly 92 (1985), 3-23), we obtain the expected value 3n/4 + O(1), from which the sequence cannot grow unlimitedly. By (Ivan Slapničar: There are no cycles in the 3n + 1 sequence; https://arxiv.org/pdf/1706.08399v1.pdf), only the trivial cycle 1-4-2-1 exists.⃞

Definition: || · ||d is the distance to the next integer.

Littlewood theorem in conventional mathematics: For all a, b ∈ κℝ, we have that:

Proof: Let r and s be the denominators of a and b with precision w and n, all natural multiples of rs. Then by the Dirichlet approximation theorem:

Refutation of the Littlewood conjecture in nonstandard mathematics: Let a = b := ⌊ω⌋-3/2. Then:

⌊ω⌋ ||⌊ω⌋a||d ||⌊ω⌋b||d = 1 ≠ 0.⃞

Prime number theorem: For x ∈ ωℝ, we have that

Proof: The asymptotic equality follows from [780], p. 76 f.⃞

Goldbach's theorem: Every n ∈ 2ℕ* \ {2} may be written as n = p + q with (p, q) ∈ ℙ2.

Proof: By the prime number theorem, the number n may be written as the sum of two primes with probability ln-2n. If we sufficiently account for statistical imprecision with another factor of ln-2n, then the claim follows, since the theorem is known to hold for n ≤ 4 1018 and 1012 ln4n < n otherwise.⃞

Similarly, Lemoine's conjecture, which was verified for n ∈ [4, 5 108]ωℕ, is proven as

Corollary: The equation 2n - 1 = p + 2q ∈ ω>6 has always solutions p, q ∈ ωℙ.⃞

Similarly, it follows directly Polignac's conjecture as

Corollary: For every fixed k ∈ κℕ*, there are O(⌊ω⌋/ln2⌊ω⌋) numbers p, p + 2k ∈ ωℙ.⃞

Cramér's gap theorem: For every p ∈ ωℙ \ {max ωℙ} and q = min ω>p, q - p = O(ln2p) holds.

Proof: Because of the prime number theorem and the basically and sectionally uniform distribution of the prime gaps, the distance q - p is at most c ⌊ln p⌋ ⌈ln p⌉ for a small c ∈ κ>0.⃞

Oppermann's conjecture follows directly as

Corollary: For every n ∈ ω>1, there is at least one (p, q) ∈ ℙ2 with n2 - n < p < n2 < q < n2 + n.⃞

The Riemann hypothesis holds as

Theorem: Every non-trivial zero of the zeta function ζ(z) is in {z ∈ ωℂ : Re(z) = ½}.

Proof: Because of [780], p. 88 ff., and [792], p. 336, we have π(x) = Li(x) + O(√x ln x).⃞