Number Theory

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

Prime number theorem: For \(\pi(x) := |\{p \in {\mathbb{P}_{\le x}} : x \in {}^{\omega}{\mathbb{R}}\}|\) holds \(\pi(\omega) = \widetilde{{_e}\omega}\omega + \mathcal{O}({_e}\omega{\omega}^{\tilde{2}})\)¹ cf. Gowers, Timothy (Ed.): The Princeton Companion to Mathematics; 1st Ed.; 2008; Princeton University Press; Princeton, p. 714 f..

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.

For induction basis \(n = 2\) resp. 3, the induction hypothesis is that the first interval contains \(x_n/{_e}x_n\) primes for \(n \in {}^{\omega}{\mathbb{N}_{\ge2}}\) and arbitrary \(x_4 \in [2, 4[\). Then the induction step from \(x_n\) to \(x_n^2\) by considering the prime gaps of prime \(p\# /q + 1\) for \(p, q \in {}^{\omega}\mathbb{P}\) proves that there are \(\pi(x_n^2) = \pi(x_n) \check{x}_n\) primes only from \(\pi(x_n) = x_n/{_e}x_n\). The average distance between the primes is \({_e}x_n\) and the maximal \(x_n^2\) to \(x_n\) behaves like \(\omega\) to \({\omega}^{\tilde{2}}.\square\)

Remark: Replacing the number 2 by \(m \in {}^{\omega }{\mathbb{N}_{>2}}\) for \(\widetilde{m}{y}^{\acute{m}}\) set-\(m\)-tuples gives the same result. The narrowly valid correction term \(\mathcal{O}({_e}\omega{\omega}^{\tilde{2}})\) disproves Legendre’s conjecture. 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 \(n \in {}^{\omega}{\mathbb{N}^{*}}\)²Scheid, Harald: Zahlentheorie; 1. Aufl.; 1991; Bibliographisches Institut; Mannheim, p. 174 f. and 354 – 365.

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, set \({a}_{m} = 1\) 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 setting i.a. \(x = \grave{y}\omega\) for \(y \in i{}^{\omega }{\mathbb{R}^{*}}.\square\)

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 \({(m, {a}_{k-1}, {a}_{k-2}, …, {a}_{1}, {a}_{0}; r, i; \#n, \&q; v, p)}_{s}\). Here \(r\) takes precedence over \(i\) and \(r = i = {a}_{0} = 0\) represents the number 0. The numerical value \(v\) has the precision \(p\). 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 >.\(\triangle\)

Remark: 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 \((0, …, 0, {a}_{k-1}, …, {a}_{0}; r, i{)}_{<}\) where the \({a}_{j}\) are finite natural numbers, gives a strict lexical well-ordering of the ANs.

Examples: The numbers \((\nu; 1, 0, 0, 0, -1{)}_{>}\) are \(1, -1, i\), and \(-i\). 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 (\(\omega, 9 \times {10}^{\omega}, 1 – {10}^{\omega})\).

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 \({a}_{k}\) of the polynomials \({a}_{m}{x}^{m} + {a}_{\acute{m}}{x}^{\acute{m}} + … + {a}_{1}x + {a}_{0}\) with \(k \in {}^{\nu}\mathbb{N}_{\le m}\) can take. 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 \({a}_{0} \ne 0\).

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

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 \(\pi(n) = \widetilde{{_e}n}n + \mathcal{O}({_e}n{n}^{\tilde{2}})\) by \(\grave{\pi}(\hat{n} + k) \le \pi(n + k) + \pi(n)\) and \(\pi(4) \le \hat{\pi}(2)\) and so on due to\[(n + k)({}_e(\hat{n}+k)-{}_e(n+k)){}_en+n({}_e(\hat{n}+k)-{}_en){}_e(n+k) \geq {}_e(\hat{n}+k){}_e(n+k){}_en.\square\]Algebraicity theorem: The numerator or denominator of every \(|z| \in {}^{\omega}\mathbb{A}_\mathbb{R} \setminus {}^{\omega}\mathbb{Q}\) is necessarily \(\mathcal{O}(\tilde{\iota}).\square\)

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 \(\kappa = \hat{n}\) the number\[\mathbb{A}(m, n) = \widetilde{\zeta(\grave{m})}\,z(m){\grave{\kappa}^m}\left( n+\mathcal{O}({_e}n) \right).\]Proof: The case \(m = 1\) requires³loc. cit., p. 323 and 330 the error term \(\mathcal{O}({_e}n n)\) and represents the number \(4{+}_{k=1}^{n}{\varphi (k)}-1\) by the \(\varphi\)-function. For \(m > 1\), the divisibility conditions neither change the error term \(\mathcal{O}({_e}n)\) nor the leading term. Polynomials or series such that gcd\(({a}_{0}, {a}_{1}, …, {a}_{m}) \ne 1\) are excluded by \(1/\zeta(\grave{m})\): The latter is given by taking the product over the primes \(p\) of all \((1 – \tilde{p}^{\grave{m}})\) absorbing here multiples of \(p\) and representing sums of GS.\(\square\)

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 \(|f(0)| \le {|f|}_{\gamma}\)⁴see a.a.O., 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.\(\square\)

Remark: In the complex case, the previous theorem teaches \(z(m) = m\). In the real case, \(z(m)\) is asymptotically equal to \({_e}m/\check{\pi} + \mathcal{O}(1)\)⁵Kac, Mark: On the Average Number of Real Roots of a Random Algebraic Equation; Bull. Amer. Math. Soc. 49 (4); 1943; 314 – 320.

Examples: For \(m = n = \acute{\nu} =: e^{\mathrm{\sigma}}\), 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 \(|{}^{\nu}\mathbb{A}_{\mathbb{C}}| = {\grave{\kappa}}^\nu\left(\check{\nu}+\mathcal{O}(\mathrm{\sigma})\right)\).

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 does not prove more than the (trivial) minimum distance between two rational numbers. Thus, the abc conjecture is wrong.

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 \(\pm 1\). The number 1 may be approximated by an real \(\omega\)-algebraic \(x\) that satisfies the polynomial or series equation \(\acute{x}x^{\acute{m}}\acute{\omega} = 1\) for \(x > 1\) or \(x^m = -\acute{x}\acute{\omega}\) for \(x < 1.\square\)

Examples: For \(m = 1\), there are \(3(n/\check{\pi})^{2}+\mathcal{O}({_e}n n)\) rational solutions. For \(m = 2\), \(\check{9}{n}^{3}/\zeta(3) + \mathcal{O}({_e}n{n}^{2})\) 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 \(z(m)\grave{\kappa}^{\acute{m}}(\kappa + \mathcal{O}({_e}\kappa))\) algebraic integer solutions.

Theorem: For every \(z \in \mathbb{Q}+ i\mathbb{Q}\) such that \(|z| \notin \{0, 1\}\), the GS is \({+}_{n=0}^{\omega}{{{z}^{n}}}=\acute{\eta}/\acute{z} \in {}^{\omega }\mathbb{T}_{\mathbb{C}}\), since the modulus of either the numerator or denominator \(\eta := {z}^{\grave{\omega}}\) is \(>{2}^{\check{\omega}}.\square\)

Theorem: Euler’s number \(e = {(1 + \tilde{\omega})}^{\omega}\) implies \(e = (k\omega + 1)/\omega!\) for \(k > \omega\) (exponential series).\(\square\)

The GPC for \(\omega\)-transcendental numbers: If a real number may be represented as an irreducible fraction \(\widetilde{ap}b \pm \tilde{s}t\), 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 \(r\) is \(\omega\)-transcendental, since the denominator \(\widetilde{aps} (bs \pm apt)\) is \(\ge \hat{p} \ge \hat{\omega} – \mathcal{O}({_e}\omega{\omega}^{\tilde{2}}) > \omega\) by the prime number theorem.\(\square \)

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-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 \(\omega\)-transcendental, since a certain power of a prime cannot be removed from numerator or denominator by simplifying.\(\square\)

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 \({+}_{n=1}^{\omega}{{\tilde{n}^{s}f(n)}\;}\) 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 \(Li_s(z) := {+}_{n=1}^{\omega}{{\tilde{n}}^s z^n}, z \in \mathbb{D}\) 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 (exact) integration of the GS (see Nonstandard Analysis) yields \(-{_e}(-\acute{x}) = Li_1(x) + \mathcal{O}(\tilde{\omega}{x}^{\grave{\omega}}/\acute{x}) + t(x){\downarrow}x\) for \(x \in [-1, 1 – \tilde{\nu}]\) and \(t(x) \in {}^{\omega }{\mathbb{R}}\) such that \(|t(x)| < {\omega}\). After applying Fermat’s (little) theorem to the numerator of \(\tilde{p}(1 – 2^{-p}\,{_2}\omega)\) for \(p = \max\, {}^{\omega}\mathbb{P}\), the GPC gives the claim.\(\square\)

Remark: If \(\omega\) is replaced by an arbitrary \(k \in {}^{\omega }\mathbb{N}_{\ge\check{\omega}}\), the preceding proof is barely more difficult. Equating \(\Gamma(\grave{n})\) and \( n!\) for \(n \in {}^{\nu}\mathbb{N}\) makes the definition below of the gamma function \(\Gamma\) worth reconsidering.

Theorem: If \(\Gamma(z)\) is defined by \(k! \, {k}^{z}/(z\grave{z} … (z + k))\) such that \(k = {\omega}^2!\) is true, \(\Gamma(z) \in \mathbb{T}_{\mathbb{R}}\) for all \(z \in R := {}^{\omega}\mathbb{Q} \setminus -{}^{\omega }\mathbb{N}\) and not too extensive rational supersets of \(R\), since there is at least one \(n\) for \(x := \Gamma(z), a_n \in \mathbb{Z}\) and \({+}_{n=0}^{\omega}{a_n{{x}^{n}}} = 0\) such that \(|a_n| > \omega.\square\)

Theorem: The BBP series \({+}_{n=1}^{\omega}{p(n)\widetilde{q(n){{b}^{n}}}}\) for \(b \in {}^{\omega }\mathbb{N}_{\ge 2}\) and integer polynomials resp. series \(p\) and \(q \in {}^{\omega }\mathbb{Z}\) with \(q(n) \ne 0\) and \(\deg(p) < \deg(q)\) only yield \(\omega\)-transcendental values by reducing the sum to a smallest common denominator \(d \ge {b}^{k} > \omega\) with \(d, k \in \mathbb{N}^{*}.\square\)

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: 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: This theorem proves the Alaoglu and Erdős 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.

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.\(\square\)

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 \(p \mid \text{gcd}(a, b)\), it follows \(p \mid c\) for \(p \in {}^{\omega} \mathbb{P}\). Otherwise it follows from \(c^k – a^m = (c^{k-s} + a^{m-r})(c^s – a^r) = c^k – a^m + a^rc^s(a^{m-\hat{r}} – c^{k-\hat{s}})\) where \(r, s \in {}^{\omega}\mathbb{Q}\) and wlog \(m \neq \hat{r}\) as well as \(k \neq \hat{s}\) (\(a^{m-\hat{r}} = c^{k-\hat{s}}\) implies equation for straight line!) \(p \mid \text{gcd}(a, c)\) resp. analogously \(p \mid \text{gcd}(b, c)\), which proves the claim.\(\square\)

Conclusion: The preceding theorem enables an infinite descent because of gcd\((a, b, c) > 1\) such that no \(n \in {}^{\omega}\mathbb{N}_{\ge 3}\) satisfies \(a^n + b^n = c^n\) for arbitrary \(a, b, c \in {}^{\omega}\mathbb{N}^{*}.\square\)

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.\(\square\)

Theorem: It holds \({+}_{n=-1}^{\omega}{\widetilde{a_n b_n}} \notin {}^{\omega}\mathbb{Q}\) only for arbitrary \(b_n \in {}^{\omega}\mathbb{N}^{*}\), if it does also for \({+}_{n=-1}^{\omega}{\widetilde{a_n}}\) or \(\widetilde{a_{-1}} – \widetilde{a_{\omega}}\) where \(a_n \in {}^{\omega}\mathbb{N}^{*}\), since \(b_n := 1\,(+\,a_n )\) (telescoping sum) may be true⁶cf. Guy, Richard K.: Unsolved Problems in Number Theory; 3rd Ed.; 2004; Springer; New York, p. 346.\(\square\)

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

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 theorem⁷Scheid, 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 \(\omega \;||\omega a|{{|}_{d}}\;||\omega b|{{|}_{d}}= 1.\square\)

Theorem: The 3\(n\) + 1 conjecture is true for every \(n \in {}^{\nu}\mathbb{N}^{*}\).

Proof: The \(m\) representations of \(n\) are from \(\{1, …, 2^mk\}\) without determining the parity of \(k \in {}^{\nu}\mathbb{N}^{*}\). Then the mean expected value is ca. \((3/4)^{\check{m}}\) of the initial value until \(k\) is determined. Cycles cannot occur⁸Slapničar, Ivan: There are no cycles in the 3n + 1 sequence, arXiv: 1706.08399v1. After \(s\) maximally forced steps, the probability for \(\grave{n} = 2^s\) to become larger is average and the claim follows by generalised iteration.\(\square\)

Example: For \(s \in {}^{(\omega)}\mathbb{C}\) where Re\((s) \le 1\) and \(z := \tilde{2}^{\acute{s}}\), \(\zeta(s) = {+}_{n=1}^{\omega}{\tilde{n}^s}\) 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:\[{\mp}_{n=1}^{\mathrm{\omega}}\tilde{n}^s=z{+}_{n=1}^{\mathrm{\check{\omega}}}\tilde{n}^s-{+}_{n=1}^{\mathrm{\omega}}\tilde{n}^s\neq\acute{z}{+}_{n=1}^{\mathrm{\omega(/2)}}\tilde{n}^s.\]Theorem: Since the Dirichlet \(L\)-function \(L\left(s,\chi\right)={+}_{n=1}^{\omega}{\chi\left(n\right)\tilde{n}^s}\) 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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