Number Theory

Number Theory
Number Theory

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

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

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 \(n = e^{\sigma} = \mathcal{O}({_e}(n!))\).

For induction basis \(n = 2\) resp. 3, the hypothesis states the first interval to contain \(x_n/{_e}x_n\) 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 \(\pi(x_n) = x_n/{_e}x_n\). The average prime gap is \({_e}x_n\), the maximal one \({_e}x_n^2\) (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.

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 \(H_n(m) \ne H_n = H_n(m, n) = m/(c + \tilde{n}) < n^{\widetilde{m}} = G_n.\square\)

Bounding theorem for \(\omega\)-transcendental numbers: Every \(z \in \mathbb{C}^{*}\) such that \(|z| \notin [\tilde{\omega}, \omega]\) is automatically \(\omega\)-transcendental.

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 \(y \in {}^{\omega }\underline{\mathbb{R}}^{*}.\square\)

Conclusions: For every \(z \in \mathbb{Q}+ \underline{\mathbb{Q}}\) where \(|z| \notin \{0, 1\}\) and \(\eta := {z}^{\grave{\omega}}\), the GS is \({\LARGE{\textbf{+}}}_{n=0}^{\omega}{{{z}^{n}}}=\acute{\eta}/\acute{z} \in {}^{\omega }\mathbb{T}_{\mathbb{C}}.\) Putting \(k = {\omega}^2!\) implies \(\Gamma(z) := k! \, {k}^{z}/(z\grave{z} … (z + k)) \in \mathbb{T}_{\mathbb{R}}\) for all \(z \in {}^{\omega}\mathbb{Q} \setminus -{}^{\omega }\mathbb{N}.\) Euler’s number \(e = {(1 + \tilde{\omega})}^{\omega}\) implies \(e = (k\omega + 1)/\omega!\) for \(k > \omega\) (exponential series).\(\square\)

Coefficient theorem for \(\omega\)-transcendental numbers: All zeros of normalised irreducible polynomials and series with at least one \({a}_{k} \notin {}^{\omega }\mathbb{Z}\) are \(\omega\)-transcendental, since they are pairwise distinct and uniquely determined as well as their non-\(\omega\)-algebraicity enforces their \(\omega\)-transcendence.\(\square\)

Definition: The notation for \(m\)-ANs is \({(m, {a}_{\acute{k}}, {a}_{k-2}, …, {a}_{1}, {a}_{0}; r, i; \#n, \&q; v, p)}_{s}\). For \(r \in {}^{\nu}\mathbb{N}^{*} (-{}^{\nu}\mathbb{N}^{*})\) exists the \(r\)-th largest (\(|r|\)-th smallest) zero with real part > 0 (< 0), where \(r = 0, i \in {}^{\nu}\mathbb{N}^{*} (-{}^{\nu}\mathbb{N}^{*})\) denotes a non-real zero with the \(i\)-th largest (\(|i|\)-th smallest) imaginary part > 0 (< 0), and the other ANs have analogous notations. The value \(\#n\) gives the quantity \(n \in {}^{\nu}\mathbb{N}^{*}\) of zeros. When at least one \({a}_{j}\) is taken as a variable, \(\&q\) gives the number \(q \in {}^{\nu}\mathbb{N}\) of repeated zeros. All \(k\)-minimal polynomials have the sign < as specification \(s\), all \(k\)-minimal series have >. For \(x \in {}^{(\omega)}\mathbb{R}\) let \({}_e^0(x) = x, {}_e^{\grave{n}}(x)={_e}({}_e^n(x))\) and \({}_e^{\grave{n}}(x) := 0\) if \({}_e^n(x)\le 1.\triangle\)

Remark: Here \(r\) takes precedence over \(i\) and \(r = i = {a}_{0} = 0\) represents the number 0. The numerical value \(v\) has the precision \(p\). Not distinguishing between repeated zeros allows the zeros of \(k\)-polynomials or \(k\)-series with integer or rational coefficients to be endowed with a strict total ordering. The information \(r, i, \#n, \&q, v, p\), and \(s\) may optionally be omitted as e.g. for rational numbers. The \((\nu+2)\)-tuple \((0, …, 0, {a}_{\acute{k}}, …, {a}_{0}; r, i{)}_{<}\) where every \({a}_{j} \in {}^{\nu}\mathbb{N}\) gives a strict lexical well-ordering of the ANs.

Remark: Let \(m \in {}^{\nu}\mathbb{N}\) be the maximum polynomial degree and \(n \in {}^{\nu}\mathbb{N}\) the maximum absolute value that the integer coefficients \({a}_{k}\) can take of the polynomials \({a}_{m}{x}^{m} + {a}_{\acute{m}}{x}^{\acute{m}} + … + {a}_{1}x + {a}_{0}\) with \(k \in {}^{\nu}\mathbb{N}_{\le m}\). This makes sense due to the symmetry of the \({a}_{k}\). The number of ANs is the number of zeros of the normalised irreducible polynomials specified by the conditions: greatest common divisor gcd of the coefficients is equal to 1, \({a}_{m} > 0\), and \({a}_{0} \ne 0\).

Examples: The numbers \((\nu; 1, 0, 0, 0, -1{)}_{>}\) are \(1, -1, \underline{1}\), and \(-\underline{1}\). The golden ratio \(\Phi := \check{1} + {5}^{\tilde{2}}/2\) may be written as \((\nu; 1, -1, -1; 1, 0; 1.618033, {10}^{-6}{)}_{<}\). The number \(0.\overline{1} = 0.1…1\) with \(\omega\) ones after the point is mid-finite and distinct from the number \(\tilde{9}\), since 9 \(\times \; 0.\overline{1} = 0.9…9 = 1 – {10}^{-\omega} \ne 1\). It is therefore \(\omega\)-transcendental and should therefore be written as (\(\omega, 9 \times {10}^{\omega}, 1 – {10}^{\omega})\).

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\[(n + k)({}_e(\hat{n}+k)-{}_e(n+k))\sigma+n({}_e(\hat{n}+k)-\sigma){}_e(n+k) \geq {}_e(\hat{n}+k){}_e(n+k)\sigma.\square\]Goldbach’s theorem: Every even whole number greater than 2 is the sum of two primes, as induction shows over all prime gaps until the maximally possible one each time.\(\square\)

Conclusion by Fortune: The previous theorem yields Bertrand’s postulate from3loc. 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 gcd\((j, m) = 1\) where \(p_n\) is the \(n\)-th prime.\(\square\)

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

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

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

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 \(m = 1\) has4loc. 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.\(\square\)

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 loc. cit., p. 160 for \(\gamma = \partial\mathbb{B}_{r}(0)\) 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 \({_e}m/\check{\pi} + \mathcal{O}(1)\)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 \({|{}^{\omega }\mathbb{Z}|}^{-m} \widetilde{{_e}\omega} \zeta(\grave{m}) \iota\).

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 \({|{}^{\omega}\mathbb{Z}|}^{-\acute{\omega}} \widetilde{{_e}\omega} \pi\). 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 rational numbers.

Theorem: For the BBP series \(s_k := {\LARGE{\textbf{+}}}_{n=1}^{\omega}{p(n)\widetilde{q(n){{b}^{n}}}}\) 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 \(\deg(p) < \deg(q)\), it holds that \(s_k \in {}^{\omega }\mathbb{T}_{\mathbb{R}}\) due to den\((s_k) \ge {b}^{m} > \omega\) with \(m \in \mathbb{N}^{*}.\square\)

Examples: For \(m = 1\), there are \(3(n/\check{\pi})^{2}+\mathcal{O}(\sigma n)\) rational 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.

Theorem: The maximum distance between two neighbouring real \(\omega\)-ANs is \(\Omega/\acute{\omega}\) for the \(\omega\)-transcendental omega constant \(\Omega = \tilde{e}^{\Omega} = W(1)\) (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 \(x < 1.\square\)

The GPC for \(\omega\)-transcendental numbers: 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 \(r \in {}^{\omega }\mathbb{T}_{\mathbb{R}}\), since the prime number theorem implies den\((\widetilde{aps} (bs \pm apt)) \ge \hat{p} \ge \hat{\omega} – \mathcal{O}({_e}\omega\;{\omega}^{\tilde{2}}) > \omega.\square\)

Theorem: It holds \(\pi \in {}^{\omega }\mathbb{T}_{\mathbb{R}}\) 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 Catalan \((G)\), Gieseking \((\pi \, {_e}\beta)\), Smarandache \(({S}_{1})\) and Taniguchi \(({C}_{T})\) are \(\omega\)-transcendental because of the GPC.\(\square\)

Theorem: The constants of \(({C}_{Artin})\), Baxter \(({C}^{2})\), Chaitin \(({\Omega}_{F})\), Champernowne \(({C}_{10})\), Copeland-Erd\H{o}s \(({C}_{CE})\) (valid for every base from \({}^{\nu}\mathbb{N}^{*}\)), Erd\H{o}s-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 \(\omega\)-transcendental, 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 positive natural numbers \(s\) and \(u\) and natural numbers \(t\), 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 rational \(|f(n)|\), the prime zeta function \(P(s)\), the polylogarithm \({Li}_{s}\) and the Lerch zeta-function \(\Phi(q, s, r)\) always yield \(\omega\)-transcendental values for rational arguments and maximal finite rational \(|q|\) 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: For \(Li_s(z) := {\LARGE{\textbf{+}}}_{n=1}^{\omega}{{\tilde{n}}^s z^n}, z \in \mathbb{D}\) and \(s \in {}^{\omega }\mathbb{C}\), let \(\gamma := Li_1(1) – {_e}\omega = {\uparrow}_{1}^{\omega}{\left( \widetilde{\left\lfloor x \right\rfloor} – \tilde{x} \right){\downarrow}x}\) where rearranging yields \(\gamma \in \; ]0, 1[\). Accepting \({_e}\omega = Li_1(\tilde{2})\;{_2}\omega\) shows \(\gamma \in {}^{\omega }\mathbb{T}_{\mathbb{R}}\) with a precision of \(\mathcal{O}(\tilde{2}^{\omega}\tilde{\omega}\;{_e}\omega)\).

Proof: The GS implies \(-{_e}(-\acute{x}) = Li_1(x) + \mathcal{O}(\tilde{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x){\downarrow}x\) 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 \(p = \max\, {}^{\omega}\mathbb{P}.\square\)

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

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

Theorem: It holds \({\LARGE{\textbf{+}}}_{n=-1}^{\omega}{\widetilde{a_n b_n}} \notin {}^{\omega}\mathbb{Q}\) 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.\(\square\)

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

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 gcd\((d, k) = 1\). 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.\(\square\)

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 \({\LARGE{\textbf{+}}}_{n=1}^{\omega}{\widetilde{a_n b_n}} \in {}^{\omega}\mathbb{T}_{\mathbb{R}}\) for \(b_n := n + 2\) resp. \(b_n := 1\).

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

Gelfond-Schneider theorem: It holds \(a^b \in {}^{\omega} \mathbb{T}_\mathbb{C}\) where \(a, c \in {}^{\omega} \mathbb{A}_\mathbb{C}^{*} \setminus \{1\}\) and infinitesimal \(\varepsilon, b \in {}^{\omega}\mathbb{A}_\mathbb{C} \setminus {}^{\omega}\mathbb{Q}\).

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

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: From \(b^n = (c^{k-r} – a^m)(c^r + 1) = c^k – a^m + c^{k-r} – a^mc^r\), it follows that \(a^m =c^{k-\hat{r}}\) and \(r \in {}^{\omega}\mathbb{Q}_{>0}\) by the previous theorem and8Walter, Wolfgang: Analysis 1; 3., verb. Aufl.; 1992; Springer; Berlin, p. 66 f., which proves the claim with gcd\((a, c) > 1\) after exponentiation.\(\square\)

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 theorem9see 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 \(\omega \;||\omega a|{{|}_{d}}\;||\omega b|{{|}_{d}}= 1.\square\)

Collatz theorem: Every \(n \in {}^{\nu}\mathbb{N}^{*}\) with \(r\) representations from \(\{1, …, 2^rm\}\) and odd or even \(m \in {}^{\nu}\mathbb{N}^{*}\) implies \(n_{\omega} \in \{1, 2, 4\}\) with start \(n_0 = n\) from \(n_{\grave{k}} := 3n_k+1-\chi_{2\mathbb{N}}(n_k)(\check{5}n_k+1)\).

Proof: The mean expected value is ca.\(\,(3/4)^{\check{m}}\) of the initial value until \(m\) is determined. Before the trivial cycle, no further can occur10Slapničar, Ivan: There are no cycles in the 3n + 1 sequence, arXiv: 1706.08399v1. After maximally \(s\) forced steps, the probability for \(\grave{n} = 2^s\) to become larger is average and the claim follows.\(\square\)

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 continuation11cf. 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.\(\square\)

© 2009-2022 by Boris Haase

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References

References
1 see Scheid, Harald: Zahlentheorie; 1. Aufl.; 1991; Bibliographisches Institut; Mannheim, p. 174 f. and 354 – 365
2 cf. 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.
3 loc. cit., p. 34 f.
4 loc. cit., p. 323 and 330
5 see loc. cit., p. 160
6 Kac, Mark: On the Average Number of Real Roots of a Random Algebraic Equation; Bull. Amer. Math. Soc. 49 (4); 1943; 314 – 320
7 cf. Guy, Richard K.: Unsolved Problems in Number Theory; 3rd Ed.; 2004; Springer; New York, p. 346
8 Walter, Wolfgang: Analysis 1; 3., verb. Aufl.; 1992; Springer; Berlin, p. 66 f.
9 see Scheid, loc. cit., p. 63
10 Slapničar, Ivan: There are no cycles in the 3n + 1 sequence, arXiv: 1706.08399v1
11 cf. Ivic, Aleksandar: The Riemann Zeta-Function; Reprint; 2003; Dover Publications; Mineola, p. 4