Number Theory

This section requires Set Theory, Topology and Nonstandard Analysis. Let \(k \in \mathbb{N}\).

Minimality theorem: For the unique \(b\)-adic expansion of every \(r \in \mathbb{R}\), min \(\{b \in \mathbb{R}_{>1}\} = 2\) holds, since the GS shows this focusing on the positions after the floating point.\(\square\)

Remark: For \(b \in \; ]1, 2[\), the development is no longer clear, as two digits (0 and 1) intended for the dual representation must be used: Restricting to one digit only makes sense for the base \(b = 1\). For \(b > 2\), uniqueness can be established because of \((1 – b^{-n})/(1 – b^{-1}) < 2\) where \(n \in \mathbb{N}\) denotes the number of decimal places of \(r\). Minimality theorem and digital computers, which are working most often within a binary system, explain the choice of 2 as basis.

Prime number theorem: For \(\pi(x) := |\{p \in {\mathbb{P}_{\le x}} : x \in {}^{\omega}{\mathbb{R}}\}|\), it holds that \(\pi(\omega) = \widetilde{{_\epsilon}\omega}\;\omega + \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}})\).

Proof: From intervals of fix length \(y \in {}^{\omega}{\mathbb{R}_{>0}}, \check{y}\) set-2-tuples of prime numbers are formed such that the first interval has the unchanged representative prime number density and the second interval is empty, then the interval with the second most prime number density is followed by the second least one etc. The Stirling formula (see Nonstandard Analysis) suggests the prime gap \(n = \epsilon^{\sigma} = \mathcal{O}({_\epsilon}(n!))\).

For induction basis \(n = 2\) resp. 3, the hypothesis states the first interval to contain \(x_n/{_\epsilon}x_n\) primes for \(n \in {}^{\omega}{\mathbb{N}_{\ge2}}\) and \(x_4 \in [2, 4[\). Stepping from \(x_n\) to \(x_n^2\) finds \(\pi(x_n^2) = \pi(x_n) \check{x}_n\) primes only from \(\pi(x_n) = x_n/{_\epsilon}x_n\). The average prime gap is \({_\epsilon}x_n\), the maximal one \({_\epsilon}x_n^2\) (see below) and the maximal \(x_n^2\) to \(x_n\) behaves like \(\omega\) to \({\omega}^{\tilde{2}}.\square\)

Remark: Replacing 2 by \(m \in {}^{\omega }{\mathbb{N}_{>2}}\) for \(\widetilde{m}{y}^{\acute{m}}\) set-\(m\)-tuples gives the same result. Induction and the sieve of Eratosthenes show by the Dirichlet prime number theorem both infinitely many prime and composite Mersenne numbers \(M_n := 2^n – 1\) for \(n \in {}^{\omega}{\mathbb{N}^{*}}\)¹see Scheid, Harald: Zahlentheorie; 1. Aufl.; 1991; Bibliographisches Institut; Mannheim, p. 174 f. and 354 – 365.

Singmaster’s theorem: There are maximally 8 distinct binomial coefficients of the same value > 1.

Proof: The existence is clear due to \(\binom{3003}{1} = \binom{78}{2} = \binom{15}{5} = \binom{14}{6}\) and the structure of Pascal’s triangle. With \(p \in {}^{\omega }{\mathbb{P}}, a,b ,c, d \in {}^{\omega }{\mathbb{N^*}}, \hat{a} \le r := p – b, \hat{a} < \hat{c} \le n := p – d, b < d\) and \(s \notin \mathbb{P}\) for every \(s \in [\max(r – \acute{a},\grave{n}), r]\), Stirling’s formula \({n!}^2\sim\pi(\hat{n}+\tilde{3}){(\tilde{\epsilon}n)}^{\hat{n}}\) and the prime number theorem imply \(\omega\binom{r}{a} \le {}_\epsilon\omega\binom{n}{c}\) for \(p \rightarrow \omega.\square\)

Giuga’s theorem: Every number \(n \in {}^{\omega }{\mathbb{N}_{\ge 2}}\) is prime if and only if \({\LARGE{\textbf{+}}}_{k=1}^{\acute{n}}{k^{\acute{n}}} \equiv -1 \mod n\).

Proof: Fermat’s (little) theorem settles the case \(n \in {}^{\omega }{\mathbb{P}} \cup {}^{\omega }{2\mathbb{N^*}}\). Otherwise, the harmonic and geometric mean \(H_n\) resp. \(G_n\) imply \({\LARGE{\textbf{+}}}_{p \in {}^{\omega}\mathbb{P}}\;{\tilde{p}} – {\LARGE{\textbf{$\times$}}}_{p \in {}^{\omega}\mathbb{P}}\;{\tilde{p}} = m/H_n – G_n^{-m} = c \in {}^{\omega }{\mathbb{N^*}}\)²cf. D. Borwein, J. M. Borwein, P. B. Borwein und R. Girgensohn: Giuga’s Conjecture on Primality; Amer. Math. Monthly 103:40-50; 1996, p. 3 f.. This contradicts \(c < 1\) due to \(H_n(m) \ne H_n = H_n(m, n) = m/(c + \tilde{n}) < n^{\widetilde{m}} = G_n.\square\)

Prime gap theorem: For the set \(M_g\) of mediate prime gaps, \(M_g \supset {}^{\omega}{2\mathbb{N}} \cup \{1\}\) holds.

Proof by induction: The claim states that beside 1 the mediate prime gaps \(m_p\) exist from 2 to \(\acute{p}\). It is true for primes \(p \in \{2, 3\}\). By stepping from \(p \rightarrow p + 2\), the prime number theorem permits no greater prime gap than those occurred as \(m_{p+2}\) . Hence \(\grave{p}\) exists, too.\(\square\)

Goldbach’s theorem: Every even whole number \(> 2\) is the sum of two primes.

Proof: For \(\hat{m} + \hat{n} = p_{m+r,n-r} + q_{m+r,n-r} + r, r \in \{0, 2, … , \max(g(n))\}\), it follows alike \(\hat{m} + \hat{n} = p_{m+s,n-s} + q_{m+s,n-s} + s,\) \(s \in \{0, 2, … , \max(g(n)) + 2\}\). This implies \(\hat{m} + \hat{n} + 2 = p_{\grave{m}+r,\grave{n}-r} + q_{\grave{m}+r,\grave{n}-r} + r, r \in \{0, 2, … , \max(g(\grave{n}))\}\). Induction yields then the claim by the previous theorem.\(\square\)

Conclusion by Fortune: The previous theorem yields Bertrand’s postulate³loc. cit., p. 34 f. for all \(n \in {}^{\omega}\mathbb{N}^*\) and min \(\{j \in {}^{\omega}\mathbb{N}_{\ge 2} : m = p_n\# \pm j \in {}^{\omega}\mathbb{P}\} = p_{\grave{n}+k}\) due to \(j \perp m\) where \(p_n\) is the \(n\)-th prime.\(\square\)

First conclusion by Hardy-Littlewood: The number of twin primes is given by \(\pi_2(n) \sim C_2\,\hat{x}\widetilde{{}_\epsilon x^2}.\square\)

Second conclusion by Hardy-Littlewood: The number of prime \(n\)-tuples for every \(n \in {}^{\omega }{\mathbb{N}_{\ge2}}\) is infinite.\(\square\)

Conclusion by Cramér: Every \(n \in {}^{\omega}\mathbb{N}_{\ge 2}\) and \(s = {\LARGE{\textbf{+}}}_{m=0}^{\omega}{{\;}_\epsilon^m(n)}\) have \(p \in {}^{\omega}\mathbb{P}\) such that \(\lfloor{\sigma}n\rfloor \le p \le \lceil{_\epsilon}s\;s\rceil.\square\)

Theorem of second Hardy-Littlewood conjecture: For \(m, n \in {}^{\omega}{\mathbb{N}_{\ge2}}\), \(\pi(m + n) \le \pi(m) + \pi(n)\) holds.

Proof by induction for \(n\): Cases \(\pi(m + \grave{n}) = \pi(m + n)\) resp. \(\pi(\grave{n}) = \grave{\pi}(n)\) and \(\pi(m + \grave{n}) = \grave{\pi}(m + n)\) are clear. For \(\pi(\grave{n}) = \pi(n)\), the last case implies the claim for \(m := n + k\) and \(\pi(n) = \widetilde{\sigma}n + \mathcal{O}(\sigma n^{\tilde{2}})\) by \(\grave{\pi}(\hat{n} + k) \le \pi(n + k) + \pi(n)\) and \(\pi(4) \le \hat{\pi}(2)\) and so on due to\[(n + k)({}_\epsilon(\hat{n}+k)-{}_\epsilon(n+k))\sigma+n({}_\epsilon(\hat{n}+k)-\sigma){}_\epsilon(n+k) \geq {}_\epsilon(\hat{n}+k){}_\epsilon(n+k)\sigma.\square\]Coefficient theorem for \(\omega\)-ANs: No zero of normalised irreducible polynomials and series with at least one \({a}_{k} \notin {}^{\omega }\mathbb{Z}\) is \(\omega\)-AN, since these are pairwise distinct and uniquely determined.\(\square\)

Bounding theorem for \(\omega\)-ANs: No \(z \in \mathbb{C}^{*}\) such that \(|z| \notin [\tilde{\omega}, \omega]\) is \(\omega\)-AN.\\ Proof: In a polynomial or series equation, put \({a}_{m} = 1\) and \({a}_{k} = -{\acute{\omega}}\) for \(k < m\), then the real case follows from the GS formula after taking the reciprocal. The exact limit value can be found by replacing \(\omega\) by \({\omega}(m) = \omega – \acute{\omega}/{\omega(m)}^{m}\). The complex case is solved by putting i.a. \(x = \grave{y}\omega\) for \(y \in {}^{\omega }{\mathbb{\underline{R}}^{*}}.\square\)

Conclusions: For every \(z \in \mathbb{R}+ \mathbb{\underline{R}}\) where \(|z| \notin \mathbb{B}\) and \(\eta := {z}^{\grave{\omega}}\), the GS is \({\LARGE{\textbf{+}}}_{n=0}^{\omega}{{{z}^{n}}}=\acute{\eta}/\acute{z} \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{C}}.\) Putting \(k = {\omega}^2!\) implies \(\Gamma(z) := k! \, {k}^{z}/(z\grave{z} … (z + k)) \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{R}}\) for all \(z \in {}^{\omega}\mathbb{R} \setminus -{}^{\omega }\mathbb{N}.\) Euler’s number \(\epsilon = {(1 + \tilde{\omega})}^{\omega}\) implies \(\epsilon = (k\omega + 1)/\omega!\) for \(k > \omega\) (exponential series).\(\square\)

Counting theorem for ANs: For the Riemann zeta function \(\zeta\) and the average number \(z(m)\) of zeros of a \(\grave{m}\)-polynomial or \(\grave{m}\)-series, the ANs asymptotically have for \(\check{\kappa} = n\) the number\[\mathbb{A}(m, n) = \widetilde{\zeta(\grave{m})}\,z(m){\grave{\kappa}^m}\left( n+\mathcal{O}( \sigma) \right).\]Proof: The case \(m = 1\) has⁴Scheid, loc. cit., p. 323 and 330 the error term \(\mathcal{O}(\sigma n)\) and represents the number \(4{\LARGE{\textbf{+}}}_{k=1}^{n}{\varphi (k)}-1\) by the \(\varphi\)-function. For \(m > 1\), the divisibility conditions neither change the error term \(\mathcal{O}( \sigma)\) nor the leading term. By \(1/\zeta(\grave{m}) = {\LARGE{\textbf{$\times$}}}_{i=1}^n(1 – \tilde{p}_i^{\grave{m}})\) (GS!), which absorbs multiples of primes \(p_i\), polynomials or series such that \(\text{ggT}({a}_{0}, {a}_{1}, \text{…} , {a}_{m}) \ne 1\) are excluded.\(\square\)

Examples: For \(m = 1\), there are \(3(n/\check{\pi})^{2}+\mathcal{O}(\sigma n)\) real solutions. For \(m = 2\), \(\check{9}{n}^{3}/\zeta(3) + \mathcal{O}(\sigma{n}^{2})\) real solutions arise, since a real polynomial of degree 2 has two real zeros with probability \({\frac{9}{16}}\) by the quadratic formula. For \({a}_{m} = 1\), there are \(z(m)\grave{\kappa}^{\acute{m}}(\kappa + \mathcal{O}(\sigma))\) algebraic integer solutions.

Fundamental theorem of algebra: Every non-constant polynomial \(p \in {}^{(\omega)}\mathbb{C}\) has at least one complex root.

Indirect proof: By performing an affine substitution of variables, reduce to the case \(\widetilde{p(0)} \ne \mathcal{O}(\iota)\). Suppose that \(p(z) \ne 0\) for all \(z \in {}^{(\omega)}\mathbb{C}\). Since \(f(z) := \widetilde{p(z)}\) is holomorphic, it holds that \(f(\tilde{\iota}) = \mathcal{O}(\iota)\). By the mean value inequality \(|f(0)| \le {|f|}_{\gamma}\)⁵see Remmert, Reinhold: Funktionentheorie 1; 3., verb. Aufl.; 1992; Springer; Berlin, p. 160 for \(\gamma = \partial^r\dot{\mathbb{C}}\) and arbitrary \(r \in {}^{(\omega)}\mathbb{R}_{>0}\), and hence \(f(0) = \mathcal{O}(\iota)\), which is a contradiction (and hence exactly \(z(m) = m\) holds).\(\square\)

Remark: In the real case, \(z(m)\) is asymptotically equal to \({_\epsilon}m/\check{\pi} + \mathcal{O}(1)\)⁶Kac, Mark: On the Average Number of Real Roots of a Random Algebraic Equation; Bull. Amer. Math. Soc. 49 (4); 1943; 314 – 320.

Conclusion: For \(m = n = \acute{\nu}\), it is true that \(|{}^{\nu}\mathbb{A}_{\mathbb{R}}| = \tilde{\pi}\sigma{\mathrm{\grave{\kappa}}^{\acute{\nu}}}\left(\hat{\nu}+\mathcal{O}(\mathrm{\sigma})\right)\) and \(|{}^{\nu}\mathbb{A}_{\mathbb{C}}| = {\grave{\kappa}}^\nu\left(\check{\nu}+\mathcal{O}(\mathrm{\sigma})\right).\square\)

Approximation theorem for \(\omega\)-ANs: The average asymptotic error to approximate every real \(\omega\)-AN of degree \(n > 1\) by a real \(\omega\)-AN of degree \(m < n\) is \({|{}^{\omega }\mathbb{Z}|}^{-m} \widetilde{{_\epsilon}\omega} \zeta(\grave{m}) \check{\pi}\).

Proof: The number of \(\omega\)-ANs approximately evenly distributed between fixed limits increases in \({}^{\omega}{\mathbb{R}}\) by a factor of approximately \(|{}^{\omega }\mathbb{Z}|\) per degree. The error corresponds to the distance between \(\omega\)-ANs. The non-real \(\omega\)-ANs are less dense.\(\square\)

Conclusion: Two distinct real \(\omega\)-ANs have an average distance of at least \({|{}^{\omega}\mathbb{Z}|}^{-\acute{\omega}} \widetilde{{_\epsilon}\omega} \pi\). Determining this minimum distance exactly requires an infinite non-linear non-convex optimisation problem to be solved. Therefore, the \(\nu\)-ANs have an approximate order of \(\mathcal{O}(\nu)\). This disproves Roth’s theorem, which only proves the minimum distance between two real numbers.\(\square\)

Theorem: The maximum distance between two neighbouring real \(\omega\)-ANs is \(\Omega/\acute{\omega}\) for the \(\omega\)-transcendental omega constant \(\Omega = \tilde{\epsilon}^{\Omega} = W(1)\) (see below Lambert-W function).

Proof: The distance between two real \(\omega\)-ANs is largest around the points \(\pm 1\). A real \(\omega\)-algebraic \(x\) approximates 1 satisfying the polynomial or series equation \(\acute{x}x^{\acute{m}}\acute{\omega} = 1\) for \(x > 1\) or \(x^m = -\acute{x}\acute{\omega}\) for \(x < 1.\square\)

Gelfond-Schneider theorem: It holds \(a^b \notin {}_{\omega}^{\omega} \mathbb{A}_\mathbb{C}\) where \(a, c \in {}^{\omega} \mathbb{A}_\mathbb{C} \setminus \mathbb{B}\) and infinitesimal \(\varepsilon, b \in {}^{\omega}\mathbb{A}_\mathbb{C} \setminus {}_{\omega}^{\omega}\mathbb{R}\).

Indirect proof: The minimal polynomials \(p\) (and \(q\)) of \(c^r\) resp. \(c^{r\pm\varepsilon} = a^b\) for maximal \(r \in {}_{\omega}^{\omega}\mathbb{R}_{>0}\) and \(f = p (\cdot q)\) lead to the contradiction \( f^\prime(c^{r(\pm\varepsilon)}) \ne 0 = (f(c^r) – f(c^{r\pm\varepsilon})) / (c^r – c^{r\pm\varepsilon}) = f^\prime(c^{r(\pm\varepsilon)}).\square\)

Theorem: The BBP series \(s_k := {\LARGE{\textbf{+}}}_{n=1}^{\omega}{p(n)\widetilde{q(n){{b}^{n}}}}\) implies, where \(b \in {}^{\omega }\mathbb{N}_{\ge 2}\) and integer polynomials resp. series \(p\) and \(q \in {}^{\omega }\mathbb{Z}\) with \(q(n) \ne 0\) and \(\deg(p) < \deg(q)\), that \(s_k \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{R}}\) due to den\((s_k) \ge {b}^{m} > \omega\) with \(m \in \mathbb{N}^{*}.\square\)

The GPC for \(\omega\)-ANs: If \(r := \widetilde{ap}b \pm \tilde{s}t \in {}^{\omega}\mathbb{R}\) is an irreducible fraction, where \(a, b, s\), and \(t\) are natural numbers, \(abst \ne 0\), \(a + s > 2\), and the (second-)greatest prime number \(p \in {}^{\omega }\mathbb{P}, p \nmid b\) and \(p \nmid s\), then \(r \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{R}}\), since the prime number theorem implies den\((\widetilde{aps} (bs \pm apt)) \ge \hat{p} \ge \hat{\omega} – \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}}) > \omega.\square\)

Theorem: The constants of Catalan \((G)\), Gieseking \((\pi \, {_\epsilon}\beta)\), Smarandache \(({S}_{1})\) and Taniguchi \(({C}_{T})\) are not \(\omega\)-algebraic because of the GPC.\(\square\)

Theorem: It holds \(\pi \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{R}}\) provided that its different representations are accepted as Wallis product, or product using the gamma function with value \(\tilde{2}\) (see Nonstandard Analysis), by alternatively applying the GPC to the Leibniz series, or the TS of arcsin\((x)\) at \(x = 1\).\(\square\)

Theorem: The constants of \(({C}_{Artin})\), Baxter \(({C}^{2})\), Chaitin \(({\Omega}_{F})\), Champernowne \(({C}_{10})\), Copeland-Erdős \(({C}_{CE})\) (valid for every base from \({}^{\nu}\mathbb{N}^{*}\)), Erdős-Borwein \((E)\), Feller-Tornier \(({C}_{FT})\), Flajolet and Richmond \((Q)\), Glaisher-Kinkelin \((A)\), Heath-Brown-Moroz \(({C}_{HBM})\), Landau-Ramanujan \((K)\), Liouville \(({£}_{Li})\), Murata \(({C}_{M})\), Pell \(({P}_{Pell})\), Prouhet-Thue-Morse \(({C}_{PTM})\), Sarnak \(({C}_{sa})\) and Stephen \(({C}_{S})\) as well as the Euler resp. Landau totient constant \((ET\) resp. \(LT)\), the twin prime constant \(({C}_{2})\) and the carefree constants \(({K}_{1}, {K}_{2}\) and \({K}_{3})\) are not \(\omega\)-ANs, since simplifying cannot remove a certain power of a prime from a fraction.\(\square\)

Theorem: The trigonometric and hyperbolic functions and their inverse functions, the digamma function \(\psi\), the Lambert-\(W\)-function, the \(Ein\) function, the (hyperbolic) sine integral \(S(h)i\), the Euler’s Beta function \(B\), and, for \(s, u \in {}^{\omega}\mathbb{N}^{*}\) and \(t \in {}^{\omega}\mathbb{N}\), the generalised error function \({E}_{t}\), the hypergeometric function \({}_{0}{F}_{t}\), the Fresnel integrals \(C\) and \(S\) and the Bessel function \({I}_{t}\) and the Bessel function of the first kind \({J}_{t}\), the Legendre function \({\chi}_{t}\), the polygamma function \({\psi}_{t}\), the generalised Mittag-Leffler function \({E}_{s,t}\), the Dirichlet series \({\LARGE{\textbf{+}}}_{n=1}^{\omega}{{\tilde{n}^{s}f(n)}\;}\) with maximally finite real \(|f(n)|\), the prime zeta function \(P(s)\), the polylogarithm \({Li}_{s}\) and the Lerch zeta-function \(\Phi(q, s, r)\) always yield no \(\omega\)-ANs for real arguments and maximal finite real \(|q|\) and \(|r|\) at points where their TS converge.

Proof: GPC, Dirichlet prime number theorem and Wallis product prove the claim. For the digamma function, the claim follows from the proof of \(\omega\)-transcendence of Euler’s constant \(\gamma\) below.\(\square\)

Theorem: Let \(\gamma := Li_1(1) – {_\epsilon}\omega = {\uparrow}_{1}^{\omega}{\left( \widetilde{\left\lfloor x \right\rfloor} – \tilde{x} \right){\downarrow}x}\) for \(Li_s(z) := {\LARGE{\textbf{+}}}_{n=1}^{\omega}{{\tilde{n}}^s z^n}, z \in {}^{1}\dot{\mathbb{C}}\) and \(s \in {}^{\omega }\mathbb{C}\), where rearranging yields \(\gamma \in \; ]0, 1[\). Accepting \({_\epsilon}\omega = Li_1(\tilde{2})\;{_2}\omega\) shows \(\gamma \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{R}}\) with a precision of \(\mathcal{O}(\tilde{2}^{\omega}\tilde{\omega}\;{_\epsilon}\omega)\).

Proof: The GS implies \(-{_\epsilon}(-\acute{x}) = Li_1(x) + \mathcal{O}(\tilde{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x){\downarrow}x\) for \(x \in [-1, 1 – \tilde{\nu}]\) and \(t(x) \in {}^{\omega }{\mathbb{R}}\) such that \(|t(x)| < {\omega}\). Apply Fermat’s (little) theorem and GPC to den\((\tilde{p}(1 – 2^{-p}\,{_2}\omega))\) for \(p = \max\, {}^{\omega}\mathbb{P}.\square\)

Theorem: It holds \({\LARGE{\textbf{+}}}_{n=-1}^{\omega}{\widetilde{a_n b_n}} \notin {}_{\omega}^{\omega}\mathbb{R}\) only for arbitrary \(b_n \in {}^{\omega}\mathbb{N}^{*}\), if it does also for \({\LARGE{\textbf{+}}}_{n=-1}^{\omega}{\widetilde{a_n}}\) or \(\widetilde{a_{-1}} – \widetilde{a_{\omega}}\) where \(a_n \in {}^{\omega}\mathbb{N}^{*}\), since \(b_n := 1\,(+\,a_n )\) (telescoping sum) may be true⁷cf. Guy, Richard K.: Unsolved Problems in Number Theory; 3rd Ed.; 2004; Springer; New York, p. 346.\(\square\)

Definition: When two numbers \(x_0, y_0 \in {}^{\omega }\mathbb{C}^{*}\) do not satisfy any non-trivial polynomial equation \(p(x, y) = 0\), so they are called \(\omega\)-algebraically independent. A real number \(\ne 0\) is said to be power-free if its modulus can only be represented as the power of a real number with integer exponent \(= \pm 1.\triangle\)

Theorem: If all \(q \in Q := {{}_{\omega}^{\omega}\mathbb{R}}_{>0}\) are power-free, \({q}^{x} \in Q\) and \({_2}\omega \gg |x| \in {}^{\omega}\mathbb{R}\), it must hold \(x \in {}^{\omega }\mathbb{Z}\).

Proof: Let wlog \(x > 0\). Since there is no contradiction for \(x \in {}^{\omega }\mathbb{N}^{*}\), assume \(x \in Q \setminus {}^{\omega }{\mathbb{N}}^{*}\). Since this implies \({q}^{x} \in {}^{\omega }{\mathbb{A}}_{R} \setminus Q\), assume \(x := k/d \in {}^{\omega }\mathbb{R}_{>0} \setminus Q\) for \(d, k \in {\mathbb{N}}^{*}\) and \(d \perp k\). This implies \({q}^{k} = {r}^{d}\) for an \(r \in Q\). The fundamental theorem of arithmetic yields a numerator or denominator of \(q\) or \(r\) greater than \(2^{\omega}\). This contradiction results in the claim.\(\square\)

Theorem: The GPC, with \(\epsilon = {(1 + \tilde{p})}^{p}\) for maximal \(p \in {}^{\omega }\mathbb{P}\) and \(\pi\) as Wallis product, yields pairwise \(\omega\)-algebraically independent representations of \(A, {C}_{2}, \gamma, \epsilon, K\) and \(\pi.\square\)

Remark: These conditions are not sufficient as the examples \(a_n := 1, b_n := 2\) resp. \((a_n ) := (12, 12, 12, 12, 12, 6, 12, 20, 30, 42, … ,\acute{\omega}\omega), b_n := 1\) show for the sums \(\check{\omega} + 1\) resp. \((\omega – 2)/\acute{\omega}\). Considering \((n!)\) resp. \((a_n)\) where \(a_n = a_n^2 – a_n + 1\), it holds that \({\LARGE{\textbf{+}}}_{n=1}^{\omega}{\widetilde{a_n b_n}} \notin {}_{\omega}^{\omega}\mathbb{A}_{\mathbb{R}}\) for \(b_n := n + 2\) resp. \(b_n := 1\).

Remark: The previous theorem proves the Alaoglu and Erdős conjecture, which states that \({p}^{x}\) and \({q}^{x}\) are \(\nu\)-real for distinct \(p, q \in {}^{\nu}\mathbb{P}\) if and only if \(x \in {}^{\nu}\mathbb{Z}\) and \(|x|\) is not excessively large.

Beal’s theorem: Equation \(a^m + b^n = c^k\) for \(a, b, c \in {}^{\omega}\mathbb{N}^{*}\) and \(k, m, n \in {}^{\omega}\mathbb{N}_{\ge 3}\) implies gcd\((a, b, c) > 1.\)

Proof: For \(b^n = (c^{kq}-a^{mr})\left(\tilde{c}^{k\acute{q}} + \tilde{a}^{m\acute{r}}\right) = c^k – a^m + c^{kq} \tilde{a}^{m\acute{r}} – \tilde{c}^{k\acute{q}} a^{mr}\), the function \(f(q,r) := c^{k(\hat{q}-1)} – a^{m(\hat{r}-1)} = 0\) is continuous in \(q, r \in {}^{\omega} \mathbb{R}_{>0}\) and \((q_0, r_0) = \left(\check{1}, \check{1}\right)\) solves the equation. Every further solution in fractions yields after exponentiation gcd\((a, c) > 1\) and thus proves the claim.\(\square\)

Conclusion: The Fermat-Catalan conjecture can be proven analogously and an infinite descent implies because of gcd\((a, b, c) > 1\) that no \(n \in {}^{\omega}\mathbb{N}_{\ge 3}\) satisfies \(a^n + b^n = c^n\) for arbitrary \(a, b, c \in {}^{\omega}\mathbb{N}^{*}.\square\)

Three-cube theorem: It holds \(S := \{n \in \mathbb{Z} : n \ne \pm 4\mod 9\} = \{n \in \mathbb{Z} : n = a^3 + b^3 + c^3 + 3(a + b)c(a – b + c)\)\(= (a + c)^3 + (b – c)^3 + c^3\} \subset a^3 + b^3 + c^3 + 6{\mathbb{Z}}\), since independent mathematical induction by equitable variables \(a, b, c \in {\mathbb{Z}}\) first shows \(\{0, \pm 1, \pm 2, \pm 3\} \subset S\), and then the claim.\(\square\)

Brocard’s theorem: It holds that \(\{(m, n) \in {}^{\omega} \mathbb{N}^2 : n! + 1 = m^2\} = \{(5, 4), (11, 5), (71, 7)\}.\)

Proof: From \(n! = \acute{m}\grave{m}\), it follows that \(m = \hat{r} \pm 1\) for \(r \in {}^{\omega} \mathbb{N}^{*}\) and \(n \ge 3\). Thus \(n! = \hat{r}(\hat{r}\pm2) = 8s(\hat{s} \pm 1)\) holds for \(s \in {}^{\omega} \mathbb{N}^{*}\). Let \(2^q \mid n!\) and \(2^{\grave{q}} \nmid n!\) for maximal \(q \in {}^{\omega} \mathbb{N}^{*}\). Therefore \(n! = 2^q(\hat{u} + 1)\) holds for \(u \in {}^{\omega} \mathbb{N}^{*}\) and necessarily \(n! = 2^q(2^{q-2} \pm 1)\). Then the prime factorisation of \(n!\) requires \(n \le 7\) giving the claim.\(\square\)

Littlewood theorem in conventional mathematics: For all \(a,b\in {}^{\nu}\mathbb{R}\) and \(n\in {}^{\nu}\mathbb{N}^{*}\), it holds that \(\underset{n\to \infty }{\mathop{\lim \inf }}\,n\;||na|{{|}_{d}}\;||nb|{{|}_{d}}=0\) where \(||\cdot|{{|}_{d}}\) is the distance to the nearest integer.

Proof: For \(k, m \in {}^{\nu}\mathbb{N}^{*}\) as denominators of the continued fraction of \(a\) resp. \(b\) with precision \(g \in {}^{\nu}{\mathbb{R}}_{>0}\) and \(n/km\) again and again integer, Dirichlet’s approximation theorem⁸see Scheid, loc. cit., p. 63 yields that: \[\underset{n\to \infty }{\mathop{\lim \inf }}\,n||na|{{|}_{d}}||nb|{{|}_{d}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,n\mathcal{O}{{(\tilde{n})}^{2}}=\underset{n\to \infty }{\mathop{\lim \inf }}\,\mathcal{O}(\tilde{n})=0.\square\]Refutation by nonstandard mathematics: For \(a = b := {\tilde{\omega}^{\check{3}}}\), it holds that \(\omega \;||\omega a|{{|}_{d}}\;||\omega b|{{|}_{d}}= 1.\square\)

Example: For \(s \in {}^{(\omega)}\mathbb{C}\) where Re\((s) \le 1\) and \(z := \tilde{2}^{\acute{s}}\), \(\zeta(s) = {\LARGE{\textbf{+}}}_{n=1}^{\omega}{\tilde{n}^s}\) has definitely no analytic continuation⁹cf. Ivic, Aleksandar: The Riemann Zeta-Function; Reprint; 2003; Dover Publications; Mineola, p. 4 and no zeros. This disproves the Riemann hypothesis:\[{\LARGE{\textbf{$\mp$}}}_{n=1}^{\mathrm{\omega}}\tilde{n}^s=z{\LARGE{\textbf{+}}}_{n=1}^{\mathrm{\check{\omega}}}\tilde{n}^s-{\LARGE{\textbf{+}}}_{n=1}^{\mathrm{\omega}}\tilde{n}^s\neq\acute{z}{\LARGE{\textbf{+}}}_{n=1}^{\mathrm{\omega(/2)}}\tilde{n}^s.\]Theorem: Since the Dirichlet \(L\)-function \(L\left(s,\chi\right)={\LARGE{\textbf{+}}}_{n=1}^{\omega}{\chi\left(n\right)\tilde{n}^s}\) has only zeros for \(s = 0\) and nontrivial Dirichlet characters \(\chi(n)\), it disproves the generalised Riemann hypothesis.\(\square\)

Example: Every \(m_0 \in {}^{(\omega)}\mathbb{N}^{*}\) implies \(m_{\tilde{\iota}} = 1\) if there are no cycles in \(m_{\grave{k}} := \left\lfloor m_k^{\check{3}-\chi_{2\mathbb{N}}\left(m_k\right)}\right\rfloor\).

Collatz theorem: Every \(n_0 \in {}^{(\omega)}\mathbb{N}^{*}\) implies \(n_{\tilde{\iota}} \in \{1, 2, 4\}\) where \(\hat{n}_{\grave{k}} := n_k + \chi_{2\mathbb{N}}(\acute{n}_k)(5n_k+2)\).

Proof: If the trivial cycle (here by ¹⁰Slapničar, Ivan: There are no cycles in the 3n + 1 sequence, arXiv: 1706.08399v1) is the only one, every such oftener descending than ascending procedure ends up below every \(n_0 \ge 2\), since the expected value is \(3^{\tilde{2}}/2\) of the previous one.\(\square\)

© 2009-2024 by Boris Haase

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