Euclidean Geometry • Equitable Distributing • Linear Optimisation • Set Theory • Nonstandardanalysis • Representations • Topology • Transcendental Numbers • Number Theory • Calculation of Times (Previous | Next)

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 ∈ ^{κ}ℝ:

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:

⃞

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^{-2}n. If we consider the statistical imprecision sufficiently with also the factor ln^{-2}n, the assertion results, since for n ≤ 4 10^{18} the correctness of the theorem is known and because of 10^{12} ln^{4}n < n otherwise.⃞

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

Corollary: For in each case fixed k ∈ ^{κ}ℕ*, we have O(⌊ω⌋/ln^{2}⌊ω⌋) numbers p, p + 2k ∈ ^{ω}ℙ.⃞

Riemann theorem: For all n ∈ ℕ_{≤100}, |ln(lcm(1, 2, …, n)) - n| ≤ √n log^{2}n 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.).⃞

© 2016 by Boris Haase

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