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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 ∈ ^{κ}ℝ, 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.⃞

Goldbach's theorem: Every even number n ∈ ℕ_{>2} may be written as the sum of two primes.

Proof: By the prime number theorem, the number n may be written as the sum of two primes with probability ln^{-2}n. If we sufficiently account for statistical imprecision with another factor of ln^{-2}n, then the claim follows, since the theorem is known to hold for n ≤ 4 10^{18} and 10^{12} ln^{4}n < n otherwise.⃞

Similarly, Lemoine's conjecture, which is true for n ∈ [4, 5 10^{8}]^{ω}ℕ, is proven as

Corollary: The equation 2n - 1 = p + 2q has always solutions p, q ∈ ^{ω}ℙ.⃞

Similarly, it follows directly Polignac's conjecture as

Corollary: For every fixed k ∈ ^{κ}ℕ*, there are O(⌊ω⌋/ln^{2}⌊ω⌋) numbers p, p + 2k ∈ ^{ω}ℙ.⃞

Riemann theorem: For x ∈ ^{ω}ℝ, we have that

Proof: The claim follows from [455], S. 335 ff.⃞

© 2016 by Boris Haase

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