Number Theory | Boris Haase's Homepage

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 $\pi(\omega) = \widetilde{{_\epsilon}\omega}\;\omega + \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}})$.

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 ${_\epsilon}x_n$, but the maximal one ${_\epsilon}x_n^2$, 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 $n \in {}^{\omega}{\mathbb{N}^{*}}$¹.

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$}}}_{p \in {}^{\omega}\mathbb{P}}\;{\tilde{p}} = m/H_n – G_n^{-m} = c \in {}^{\omega }{\mathbb{N^*}}$². 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.$\square$

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 $m = 1$ has³ 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: An affine substitution of variables reduces to the case $\widetilde{p(0)} \ne \mathcal{O}(\iota)$. Then $f(z) := \widetilde{p(z)}$ is holomorphic for all $z \in {}^{(\omega)}\mathbb{C}$ by supposing $p(z) \ne 0$. For $f(\tilde{\iota}) = \mathcal{O}(\iota)$, the mean value inequality (see Nonstandard Analysis) yields $|f(0)| \le {|f|}_{\gamma}$ where $\gamma = \partial^r\dot{\mathbb{C}}$ and arbitrary $r \in {}^{(\omega)}\mathbb{R}_{>0}$, and hence $f(0) = \mathcal{O}(\iota).\square$

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.

Remark: In the real case, $z(m)$ is asymptotically equal to ${_\epsilon}m/\check{\pi} + \mathcal{O}(1)$⁴.

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 $|{}^{\nu}\mathbb{A}_{\mathbb{C}}| = {\grave{\kappa}}^\nu\left(\check{\nu}+\mathcal{O}(\mathrm{\sigma})\right).\square$

Bunyakovsky theorem: If $a_n > 0$ holds for a minimal polynomial $p \in {}^{(\omega)}\mathbb{C}$ of degree $n$ where $p(i) \perp p(j)$ for every ${}^{\omega }\mathbb{N}^* \owns i < j \in {}^{\omega }\mathbb{N}^*$, then it has infinitely many primes as values.

Proof: The Dirichlet prime number theorem yields for $m \in {}^{\omega }\mathbb{N}^*$ infinitely many primes $q_m = b m + c$ where $b \perp c \in {}^{\omega }\mathbb{N}^*$ are fix. Vice versa there are infinitely many $m$ such that $d_m := p(m) = (q_m – c)/m$. By rearranging the latter formula, the claim follows by mathematical induction for the polynomial degree $n.\square$

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 $m \in \mathbb{N}^{*}.\square$

Theorem: The maximum distance between two neighbouring real $\omega$-ANs is $\Omega/\acute{\omega}$ for the not $\omega$-algebraic 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 $x < 1.\square$

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 begins with false assumptions.$\square$

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$(\widetilde{aps} (bs \pm apt)) \ge \hat{p} \ge \hat{\omega} – \mathcal{O}({_\epsilon}\omega\;{\omega}^{\tilde{2}}) > \omega.\square$

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 $f = p (q)$ lead to the contradiction ${}^1f(c^{r(\pm\varepsilon)}) \ne 0 = (f(c^r) – f(c^{r\pm\varepsilon})) / (c^r – c^{r\pm\varepsilon}) = {}^1f(c^{r(\pm\varepsilon)}).\square$

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 missing of the $\omega$-algebraicity 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 true⁵.$\square$

Theorem: All $p \in {}^{\omega }\mathbb{P}$ where $q \in Q := {{}_{\omega}^{\omega}\mathbb{R}}_{>0} \owns {q}^{x}$ and ${_2}\omega \gg |x| \in {}^{\omega}\mathbb{R}$ imply $x \in {}^{\omega }\mathbb{Z}$ for $p^2 \nmid \text{num}(q)\text{den}(q)$.

Indirect 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 $2^{\omega}.\square$

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.

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 ⁶) 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.$\square$

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.$\triangle$

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 $\acute{a}_{\grave{n}} = a_n \grave{a}_n$, 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. $b_n := 1$.

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

Indirect proof: Put necessarily $(da)^n + (db)^m = (dc)^n$ to cancel $d$ after computing mod $d$. Multiply $b^s + e^m = c^s$ generalised by $d^{rm}$ again to yield $b = 1$, $c = d$ and $s \ne m \notin 2\mathbb{N}$. The second general form $1 + d^s = e^m$ shows likewise the claim by contradiction for $m \ne s \notin 2\mathbb{N}$.$\square$

Conclusion: Due to $m \ne s$, no $n \in {}^{\omega}\mathbb{N}_{\ge 3}$ satisfies $a^n + b^n = c^n$ for arbitrary $a, b, c \in {}^{\omega}\mathbb{N}^{*}.\square$

Catalan’s theorem: It holds that $\{(m, n, x, y) \in {}^{\omega}\mathbb{N}_{\ge 2}^4 : 1 + x^m = y^n\} = \{(3, 2, 2, 3)\}$.

Indirect proof: The previous theorem gives min$(m, n) = 2$. Squaring shows $1 + 4{\check{x}}^2 = (1 + \acute{y})^n$ and $(1 + 2^rz)^n \equiv 1 + 2^r{\check{x}}^2 \equiv 1$ mod $2^{\grave{r}}$ for every $r \in {}^{\omega}\mathbb{N}_{\ge 3}$, and $1 + x^2 \equiv 2 \ne 0 \equiv 2^n{\check{y}}^n$ mod $8$ where $\acute{y} = 4z$ and $n$ is odd. Odd $m$ and $s \in {}^{\omega}\mathbb{N}^{*}$ imply $x^m = \acute{y}\grave{y}$ where $s^m \ne \acute{y} \in {}^{\omega}2\mathbb{N}^{*}$ such that $x^m = 8\check{s}\acute{s} = 8.\square$

Erdős-Moser theorem: Faulhaber’s formula (see Nonstandard Analysis) implies for $k, n \in {}^{\omega} \mathbb{N}^{*}$ that from $n\grave{n} \mid {\LARGE{\textbf{+}}}_{m=1}^{n} {m^k} = \grave{n}^k$ only the solution $k = 1 = \acute{n}$ follows due to $1 < n \nmid \grave{n}.\square$

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.$\square$

Wilson’s prime number theorem: Only the primes $p \in \{5, 13, 563\}$ satisfy $q := p^2 \mid (\acute{p}! + 1)$.

Indirect proof: Adding and subtracting afterwards a suitable power of two on the left-hand side, division by $\acute{p}$ or $\grave{p}$ yields in $\hat{n}q + \acute{q} = \acute{p}!$ for $n \in {}^{\omega}\mathbb{N}^{*}$ a higher one on the right-hand side for sufficiently big $p.\square$

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 theorem⁷ 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 continuation⁸ 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.$\square$

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References

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