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Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlgsfcl3 21101* Closure of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4a 21102* Same as lgsval4 21100 for positive . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsneg 21103 The Legendre symbol is either even or odd under negation with respect to the second parameter according to the sign of the first. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsneg1 21104 The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsmod 21105 The Legendre (Jacobi) symbol is preserved under reduction when is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdilem 21106 Lemma for lgsdi 21116 and lgsdir 21114: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem1 21107 Lemma for lgsdir2 21112. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem2 21108 Lemma for lgsdir2 21112. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem3 21109 Lemma for lgsdir2 21112. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem4 21110 Lemma for lgsdir2 21112. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem5 21111 Lemma for lgsdir2 21112. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2 21112 The Legendre symbol is completely multiplicative at . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdirprm 21113 The Legendre symbol is completely multiplicative at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir 21114 The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 21130 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdilem2 21115* Lemma for lgsdi 21116. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdi 21116 The Legendre symbol is completely multiplicative in its right argument. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgsne0 21117 The Legendre symbol is nonzero (and hence equal to or ) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgsabs1 21118 The Legendre symbol is nonzero (and hence equal to or ) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgssq 21119 The Legendre symbol at a square is equal to . Together with lgsmod 21105 this implies that the Legendre symbol takes value at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgssq2 21120 The Legendre symbol at a square is equal to . (Contributed by Mario Carneiro, 5-Feb-2015.)

Theorem1lgs 21121 The Legendre symbol at . (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgs1 21122 The Legendre symbol at . (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgsdirnn0 21123 Variation on lgsdir 21114 valid for all but only for positive . (The exact location of the failure of this law is for , , in which case but .) (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgsdinn0 21124 Variation on lgsdi 21116 valid for all but only for positive . (The exact location of the failure of this law is for , , and some in which case but when is not a quadratic residue mod .) (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgsqrlem1 21125 Lemma for lgsqr 21130. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem2 21126* Lemma for lgsqr 21130. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem3 21127* Lemma for lgsqr 21130. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem4 21128* Lemma for lgsqr 21130. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem5 21129* Lemma for lgsqr 21130. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsqr 21130* The Legendre symbol for odd primes is iff the number is not a multiple of the prime (in which case it is , see lgsne0 21117) and the number is a quadratic residue (it is for nonresidues by the process of elimination from lgsabs1 21118). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsdchrval 21131* The Legendre symbol function , where is an odd positive number, is a Dirichlet character modulo . (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                     RHom

Theoremlgsdchr 21132* The Legendre symbol function , where is an odd positive number, is a real Dirichlet character modulo . (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                     RHom

Theoremlgseisenlem1 21133* Lemma for lgseisen 21137. If and , then for any even , is also an even integer . To simplify these statements, we divide all the even numbers by , so that it becomes the statement that is an integer between and . (Contributed by Mario Carneiro, 17-Jun-2015.)

Theoremlgseisenlem2 21134* Lemma for lgseisen 21137. The function is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)

Theoremlgseisenlem3 21135* Lemma for lgseisen 21137. (Contributed by Mario Carneiro, 17-Jun-2015.)
ℤ/n       mulGrp       RHom       g

Theoremlgseisenlem4 21136* Lemma for lgseisen 21137. The function is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 18-Jun-2015.)
ℤ/n       mulGrp       RHom

Theoremlgseisen 21137* Eisenstein's lemma, an expression for when are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquadlem1 21138* Lemma for lgsquad 21141. Count the members of with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquadlem2 21139* Lemma for lgsquad 21141. Count the members of with even coordinates, and combine with lgsquadlem1 21138 to get the total count of lattice points in (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquad 21141 The Law of Quadratic Reciprocity. If and are distinct odd primes, then the product of the Legendre symbols and is the parity of . This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad2 21144 Extend lgsquad 21141 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad3 21145 Extend lgsquad2 21144 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremm1lgs 21146 The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime iff mod 4. (Contributed by Mario Carneiro, 19-Jun-2015.)

13.4.10  All primes 4n+1 are the sum of two squares

Theorem2sqlem1 21147* Lemma for 2sq 21160. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem2 21148* Lemma for 2sq 21160. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremmul2sq 21149 Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem3 21150 Lemma for 2sqlem5 21152. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem4 21151 Lemma for 2sqlem5 21152. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem5 21152 Lemma for 2sq 21160. If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem6 21153* Lemma for 2sq 21160. If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem7 21154* Lemma for 2sq 21160. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem8a 21155* Lemma for 2sqlem8 21156. (Contributed by Mario Carneiro, 4-Jun-2016.)

Theorem2sqlem8 21156* Lemma for 2sq 21160. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem9 21157* Lemma for 2sq 21160. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem10 21158* Lemma for 2sq 21160. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem11 21159* Lemma for 2sq 21160. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sq 21160* All primes of the form are sums of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqblem 21161 The converse to 2sq 21160. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqb 21162* The converse to 2sq 21160. (Contributed by Mario Carneiro, 20-Jun-2015.)

13.4.11  Chebyshev's Weak Prime Number Theorem, Dirichlet's Theorem

Theoremchebbnd1lem1 21163 Lemma for chebbnd1 21166: show a lower bound on π at even integers using similar techniques to those used to prove bpos 21077. (Note that the expression is actually equal to , but proving that is not necessary for the proof, and it's too much work.) The key to the proof is bposlem1 21068, which shows that each term in the expansion is at most , so that the sum really only has nonzero elements up to , and since each term is at most , after taking logs we get the inequality π , and bclbnd 21064 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
π

Theoremchebbnd1lem2 21164 Lemma for chebbnd1 21166: Show that does not change too much between and . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd1lem3 21165 Lemma for chebbnd1 21166: get a lower bound on π that is independent of . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchebbnd1 21166 The Chebyshev bound: The function π is eventually lower bounded by a positive constant times . Alternatively stated, the function π is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilimlem1 21167 Lemma for chtppilim 21169. (Contributed by Mario Carneiro, 22-Sep-2014.)
π        π

Theoremchtppilimlem2 21168* Lemma for chtppilim 21169. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilim 21169 The function is asymptotic to π, so it is sufficient to prove to establish the PNT. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchto1ub 21170 The function is upper bounded by a linear term. Corollary of chtub 20996. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd2 21171 The Chebyshev bound, part 2: The function π is eventually upper bounded by a positive constant times . Alternatively stated, the function π is eventually bounded. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchto1lb 21172 The function is lower bounded by a linear term. Corollary of chebbnd1 21166. (Contributed by Mario Carneiro, 8-Apr-2016.)

Theoremchpchtlim 21173 The ψ and functions are asymptotic to each other, so is sufficient to prove either or ψ to establish the PNT. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpo1ub 21174 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ

Theoremchpo1ubb 21175* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
ψ

Theoremvmadivsum 21176* The sum of the von Mangoldt function over is asymptotic to . Equation 9.2.13 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 16-Apr-2016.)
Λ

Theoremvmadivsumb 21177* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremrplogsumlem1 21178* Lemma for rplogsum 21221. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremrplogsumlem2 21179* Lemma for rplogsum 21221. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Λ

Theoremdchrisum0lem1a 21180 Lemma for dchrisum0lem1 21210. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremrpvmasumlem 21181* Lemma for rpvmasum 21220. Calculate the "trivial case" estimate Λ , where is the principal Dirichlet character. Equation 9.4.7 of [Shapiro], p. 376. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                     Λ

Theoremdchrisumlema 21182* Lemma for dchrisum 21186. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumlem1 21183* Lemma for dchrisum 21186. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^        ..^

Theoremdchrisumlem2 21184* Lemma for dchrisum 21186. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisumlem3 21185* Lemma for dchrisum 21186. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisum 21186* If is a positive decreasing function approaching zero, then the infinite sum is convergent, with the partial sum within of the limit . Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusumlema 21187* Lemma for dchrmusum 21218 and dchrisumn0 21215. Apply dchrisum 21186 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 21188* The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by , is bounded, provided that . Lemma 9.4.2 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem1 21189* An alternative expression for a Dirichlet-weighted von Mangoldt sum in terms of the Möbius function. Equation 9.4.11 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr                                          Λ

Theoremdchrvmasum2lem 21190* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 21189. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 21191* Combine the results of dchrvmasumlem1 21189 and dchrvmasum2lem 21190 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 21192* Lemma for dchrvmasum 21219. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 21193* Lemma for dchrvmasum 21219. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 21194* Lemma for dchrvmasum 21219 and dchrvmasumif 21197. Apply dchrisum 21186 for the function , which is decreasing above (or above 3, the nearest integer bound). (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem1 21195* Lemma for dchrvmasumif 21197. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 21196* Lemma for dchrvmasum 21219. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 21197* An asymptotic approximation for the sum of Λ conditional on the value of the infinite sum . (We will later show that the case is impossible, and hence establish dchrvmasum 21219.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

Theoremdchrvmaeq0 21198* The set is the collection of all non-principal Dirichlet characters such that the sum is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fval 21199* Value of the function , the divisor sum of a Dirichlet character. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisum0fmul 21200* The function , the divisor sum of a Dirichlet character, is a multiplicative function (but not completely multiplicative). Equation 9.4.27 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

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