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| | __NOTOC__ | | __NOTOC__ |
| − | = Welcome to MWiki = | + | = Willkommen bei MWiki = |
| − | == Theorem of the month == | + | == Satz des Monats == |
| − | === Fermat's Last Theorem === | + | === Counting theorem for algebraic numbers === |
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| − | For all <math>p \in {}^{\omega }{\mathbb{P}_{\ge 3}}</math> and <math>x, y, z \in {}^{\omega }{\mathbb{N}^{*}}</math>, always <math>x^p + y^p \ne z^p</math> holds and thus for all <math>m \in {}^{\omega }{\mathbb{N}_{\ge 3}}</math> instead of <math>p</math>.
| + | The number <math>\mathbb{A}(m, n)</math> of algebraic numbers of polynomial or series degree <math>m</math> and thus in general for the Riemann zeta function <math>\zeta</math> asymptotically satisfies the equation\[\mathbb{A}(m, n) = \widehat{\zeta(\grave{m})}\,z(m){{(2n+1)}^{m}}\left( n+\mathcal{O}({_e}n) \right),\]where <math>z(m)</math> is the average number of zeros of a polynomial or series. |
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| | ==== Proof: ==== | | ==== Proof: ==== |
| − | Because of [[w:Fermat's little theorem|<span class="wikipedia">Fermat's little theorem</span>]], rewritten, <math>f_{akp}(n) := (2n + a - kp)^p - n^p - (n + a)^p \ne 0</math> is to show for <math>a, k, n \in {}^{\omega }{\mathbb{N}^{*}}</math> where <math>kp < n</math>.
| + | The case <math>m = 1</math> requires by <ref name="Scheid">[[w:Harald Scheid|<span class="wikipedia">Scheid, Harald</span>]]: ''Zahlentheorie'' : 1. Aufl.; 1991; Bibliographisches Institut; Mannheim; ISBN 9783411148417, S. 323.</ref> the error term <math>\mathcal{O}({_e}n n)</math> and represents the number <math>4\sum\limits_{k=1}^{n}{\varphi (k)}-1</math> by the <math>\varphi</math>-function. For <math>m > 1</math>, the divisibility conditions neither change the error term <math>\mathcal{O}({_e}n)</math> nor the leading term. Polynomials or series such that gcd<math>({a}_{0}, {a}_{1}, ..., {a}_{m}) \ne 1</math> are excluded by <math>1/\zeta(\grave{m})</math>: The latter is given by taking the product over the prime numbers <math>p</math> of all <math>(1 - {p}^{-\grave{m}})</math> absorbing here multiples of <math>p</math> and representing sums of geometric series.<math>\square</math> |
| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:100%; overflow:auto;">
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| − | <div style="font-weight:bold;line-height:1.6;">Proof details</div>
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| − | <div class="mw-collapsible-content">From <math>x := n, y:= n + a</math> and <math>z := 2n + a + d</math> where <math>d \in {}^{\omega }{\mathbb{N}^{*}}</math>, it follows due to <math>z^p \equiv y, y^p \equiv y</math> and <math>z^p \equiv z</math> first <math>d \equiv 0 \mod p</math>, then <math>d = \pm kp</math>. Since <math>x + y = 2n + a > z</math> is required, <math>f_{akp}(n)</math> is chosen properly.</div></div>
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| − | [[w:Mathematical induction|<span class="wikipedia">Induction</span>]] for <math>n</math> implies the claim due to the case <math>m = 4</math><ref name="Ribenboim">[[w:Paulo Ribenboim|<span class="wikipedia">Ribenboim, Paulo</span>]]: ''Thirteen Lectures on Fermat's Last Theorem'' : 1979; Springer; New York; ISBN 9780387904320, p. 35 - 38.</ref> and <math>y > x > p</math><ref name="loccit">loc. cit., p. 226.</ref>:
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| − | '''Induction basis''' <math>(n \le p): f_{akp}(n) \ne 0</math> for all <math>a, k</math> and <math>p</math>. Let <math>r \in {}^{\omega }{\mathbb{N}_{< p}}</math>.
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| − | '''Induction step''' <math>\,(n = q + r \; \rightarrow \; n^{*} = n + p):</math> Let <math>f_{akp}(n^{*}) \ge 0</math>, but <math>f_{akp}(n) < 0</math>, since <math>f_{akp}(n)</math> is [[w:Monotonic function|<span class="wikipedia">strictly monotonically increasing</span>]] and otherwise nothing to prove.
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:100%; overflow:auto;">
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| − | <div style="font-weight:bold;line-height:1.6;">Proof details</div>
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| − | <div class="mw-collapsible-content">The strict monotonicity follows from (continuously) differentiating by <math>n</math> such that <math>f_{akp}(n)' = p(2(2n + a - kp)^{p - 1} - n^{p - 1} - (n + a)^{p - 1}) > 0</math>.</div></div>
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| − | It holds <math>f_{akp}(n^{*}) = (\int_0^{n^{*}}{f_{akp}(v)}dv)' \ne 0</math>, since <math>(n^{*})^{p + 1} + (n^{*} + a)^{p + 1}</math> does not divide <math>((n^{*})^p + (n^{*} + a)^p)^2</math> after separating the positive factor as [[w:Polynomial long division|<span class="wikipedia">polynomial division</span>]] shows.<math>\square</math>
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| − | <div class="toccolours mw-collapsible mw-collapsed" style="width:100%; overflow:auto;">
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| − | <div style="font-weight:bold;line-height:1.6;">Proof details</div>
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| − | <div class="mw-collapsible-content"><math>\int_0^{n^{*}}{f_{akp}(v)}dv = ((2n^{*} + a - kp)^{p + 1} / 2 - (n^{*})^{p + 1} - (n^{*} + a)^{p + 1})/(p + 1) + t = ((2n^{*} + a - kp)^{(p + 1)/2} \pm \sqrt{2(n^{*})^{p + 1} + 2(n^{*} + a)^{p + 1}})^2/(2p + 2) + t</math> for <math>t \in {}^{\omega}{\mathbb{Q}}</math> where the third binomial formula <math>r^2 - s^2 = (r \pm s)^2 := (r + s)(r - s)</math> was used. After separating the negligible factor <math>\hat{2}((2n^{*} + a - kp)^{(p + 1)/2} + \sqrt{2(n^{*})^{p + 1} + 2(n^{*} + a)^{p + 1}})/(p + 1)</math>, the derivative is just <math>(\hat{2}(2n^{*} + a - kp)^{(p - 1)/2} - \hat{2}((n^{*})^p + (n^{*} + a)^p)/\sqrt{2(n^{*})^{p + 1} + 2(n^{*} + a)^{p + 1}})</math>. After squaring the terms, the polynomial division gives <math>(n^{*})^{p - 1} + (n^{*} + a)^{p - 1} + a^2(n^{*})^{p - 1}(n^{*} + a)^{p - 1}/((n^{*})^{p + 1} + (n^{*} + a)^{p + 1})</math> as recalculating by multiplication confirms.</div></div>
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| | == Recommended reading == | | == Recommended reading == |
Willkommen bei MWiki
Satz des Monats
Counting theorem for algebraic numbers
The number [math]\displaystyle{ \mathbb{A}(m, n) }[/math] of algebraic numbers of polynomial or series degree [math]\displaystyle{ m }[/math] and thus in general for the Riemann zeta function [math]\displaystyle{ \zeta }[/math] asymptotically satisfies the equation\[\mathbb{A}(m, n) = \widehat{\zeta(\grave{m})}\,z(m){{(2n+1)}^{m}}\left( n+\mathcal{O}({_e}n) \right),\]where [math]\displaystyle{ z(m) }[/math] is the average number of zeros of a polynomial or series.
Proof:
The case [math]\displaystyle{ m = 1 }[/math] requires by [1] the error term [math]\displaystyle{ \mathcal{O}({_e}n n) }[/math] and represents the number [math]\displaystyle{ 4\sum\limits_{k=1}^{n}{\varphi (k)}-1 }[/math] by the [math]\displaystyle{ \varphi }[/math]-function. For [math]\displaystyle{ m \gt 1 }[/math], the divisibility conditions neither change the error term [math]\displaystyle{ \mathcal{O}({_e}n) }[/math] nor the leading term. Polynomials or series such that gcd[math]\displaystyle{ ({a}_{0}, {a}_{1}, ..., {a}_{m}) \ne 1 }[/math] are excluded by [math]\displaystyle{ 1/\zeta(\grave{m}) }[/math]: The latter is given by taking the product over the prime numbers [math]\displaystyle{ p }[/math] of all [math]\displaystyle{ (1 - {p}^{-\grave{m}}) }[/math] absorbing here multiples of [math]\displaystyle{ p }[/math] and representing sums of geometric series.[math]\displaystyle{ \square }[/math]
Recommended reading
Nonstandard Mathematics
References