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Number Theory

Number Theory

In the following, the results of set theory are presupposed.

Collatz theorem: The two sequences n/2 for even n and 3n ± 1 for odd n end for n ∈ ℕ* always in a cycle.

Proof: We want to exclude unlimited growth of the sequences. Iterating once, we obtain according to (Jeffrey C. Lagarias, "The 3x+1 problem and its generalizations", American Mathematical Monthly 92 (1985), 3-23) the expectation value 3n/4 ± O(1) and thereof the assertion.⃞

Littlewood theorem in conventional mathematics: We have for all a, b ∈ κℝ:

formula_001

Proof: Let be r resp. s the denominators of the continued fraction of a resp. b with precision w and n again and again a natural multiple of rs. Then we have according to Dirichlet's approximation theorem:

formula_002

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

⌊ω⌋ ||⌊ω⌋a||d ||⌊ω⌋b||d = 1 ≠ 0.⃞

Goldbach theorem: Each even number n ∈ ℕ>2 can be represented as sum of two primes.

Proof: The number n is sum of two primes according to the prime number theorem with the probability ln-2n. If we consider the statistical imprecision sufficiently with also the factor ln-2n, the assertion results, since for n ≤ 4 1018 the correctness of the theorem is known and because of 1012 ln4n < n otherwise.⃞

From the reason just mentioned, it follows directly the Polignac conjecture as

Corollary: For in each case fixed k ∈ κℕ*, we have O(⌊ω⌋/ln2⌊ω⌋) numbers p, p + 2k ∈ ωℙ.⃞

Riemann theorem: For all n ∈ ℕ≤100, |ln(lcm(1, 2, , n)) - n| ≤ √n log2n is true.

Proof: From computer calculations and the prime number theorem, it follows ln(lcm(1, 2, , n)) ~ n (cf. [455], pp. 337 f. and [792], pp. 339 f.).⃞

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