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Theorem List for Metamath Proof Explorer - 20501-20600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlgsval3 20501 The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsvalmod 20502 The Legendre symbol is equivalent to , . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4 20503* Restate lgsval 20487 for nonzero , where the function has been abbreviated into a self-referential expression taking the value of on the primes as given. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfcl3 20504* Closure of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4a 20505* Same as lgsval4 20503 for positive . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsneg 20506 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 20507 The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsmod 20508 The Legendre (Jacobi) symbol is preserved under reduction when is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdilem 20509 Lemma for lgsdi 20519 and lgsdir 20517: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem1 20510 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem2 20511 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem3 20512 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem4 20513 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem5 20514 Lemma for lgsdir2 20515. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2 20515 The Legendre symbol is completely multiplicative at . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdirprm 20516 The Legendre symbol is completely multiplicative at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir 20517 The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 20533 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 20518* Lemma for lgsdi 20519. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdi 20519 The Legendre symbol is completely multiplicative in its right argument. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgsne0 20520 The Legendre symbol is nonzero (and hence equal to or ) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgsabs1 20521 The Legendre symbol is nonzero (and hence equal to or ) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgssq 20522 The Legendre symbol at a square is equal to . Together with lgsmod 20508 this implies that the Legendre symbol takes value at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgssq2 20523 The Legendre symbol at a square is equal to . (Contributed by Mario Carneiro, 5-Feb-2015.)

Theorem1lgs 20524 The Legendre symbol at . (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgs1 20525 The Legendre symbol at . (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgsdirnn0 20526 Variation on lgsdir 20517 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 20527 Variation on lgsdi 20519 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 20528 Lemma for lgsqr 20533. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem2 20529* Lemma for lgsqr 20533. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem3 20530* Lemma for lgsqr 20533. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem4 20531* Lemma for lgsqr 20533. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem5 20532* Lemma for lgsqr 20533. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsqr 20533* The Legendre symbol for odd primes is iff the number is not a multiple of the prime (in which case it is , see lgsne0 20520) and the number is a quadratic residue (it is for nonresidues by the process of elimination from lgsabs1 20521). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsdchrval 20534* 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 20535* 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 20536* Lemma for lgseisen 20540. 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 20537* Lemma for lgseisen 20540. The function is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)

Theoremlgseisenlem3 20538* Lemma for lgseisen 20540. (Contributed by Mario Carneiro, 17-Jun-2015.)
ℤ/n       mulGrp       RHom       g

Theoremlgseisenlem4 20539* Lemma for lgseisen 20540. 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 20540* Eisenstein's lemma, an expression for when are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquadlem1 20541* Lemma for lgsquad 20544. Count the members of with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquadlem2 20542* Lemma for lgsquad 20544. Count the members of with even coordinates, and combine with lgsquadlem1 20541 to get the total count of lattice points in (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquad 20544 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 20547 Extend lgsquad 20544 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad3 20548 Extend lgsquad2 20547 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremm1lgs 20549 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 20550* Lemma for 2sq 20563. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem2 20551* Lemma for 2sq 20563. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremmul2sq 20552 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 20553 Lemma for 2sqlem5 20555. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem4 20554 Lemma for 2sqlem5 20555. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem5 20555 Lemma for 2sq 20563. 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 20556* Lemma for 2sq 20563. 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 20557* Lemma for 2sq 20563. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem8a 20558* Lemma for 2sqlem8 20559. (Contributed by Mario Carneiro, 4-Jun-2016.)

Theorem2sqlem8 20559* Lemma for 2sq 20563. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem9 20560* Lemma for 2sq 20563. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem10 20561* Lemma for 2sq 20563. 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 20562* Lemma for 2sq 20563. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sq 20563* All primes of the form are sums of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqblem 20564 The converse to 2sq 20563. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqb 20565* The converse to 2sq 20563. (Contributed by Mario Carneiro, 20-Jun-2015.)

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

Theoremchebbnd1lem1 20566 Lemma for chebbnd1 20569: show a lower bound on π at even integers using similar techniques to those used to prove bpos 20480. (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 20471, 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 20467 finishes the proof. (Contributed by Mario Carneiro, 22-Sep-2014.) (Revised by Mario Carneiro, 15-Apr-2016.)
π

Theoremchebbnd1lem2 20567 Lemma for chebbnd1 20569: Show that does not change too much between and . (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd1lem3 20568 Lemma for chebbnd1 20569: get a lower bound on π that is independent of . (Contributed by Mario Carneiro, 21-Sep-2014.)
π

Theoremchebbnd1 20569 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 20570 Lemma for chtppilim 20572. (Contributed by Mario Carneiro, 22-Sep-2014.)
π        π

Theoremchtppilimlem2 20571* Lemma for chtppilim 20572. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtppilim 20572 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 20573 The function is upper bounded by a linear term. Corollary of chtub 20399. (Contributed by Mario Carneiro, 22-Sep-2014.)

Theoremchebbnd2 20574 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 20575 The function is lower bounded by a linear term. Corollary of chebbnd1 20569. (Contributed by Mario Carneiro, 8-Apr-2016.)

Theoremchpchtlim 20576 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 20577 The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 16-Apr-2016.)
ψ

Theoremchpo1ubb 20578* The ψ function is upper bounded by a linear term. (Contributed by Mario Carneiro, 31-May-2016.)
ψ

Theoremvmadivsum 20579* 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 20580* Give a total bound on the von Mangoldt sum. (Contributed by Mario Carneiro, 30-May-2016.)
Λ

Theoremrplogsumlem1 20581* Lemma for rplogsum 20624. (Contributed by Mario Carneiro, 2-May-2016.)

Theoremrplogsumlem2 20582* Lemma for rplogsum 20624. Equation 9.2.14 of [Shapiro], p. 331. (Contributed by Mario Carneiro, 2-May-2016.)
Λ

Theoremdchrisum0lem1a 20583 Lemma for dchrisum0lem1 20613. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremrpvmasumlem 20584* Lemma for rpvmasum 20623. 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 20585* Lemma for dchrisum 20589. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrisumlem1 20586* Lemma for dchrisum 20589. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^        ..^

Theoremdchrisumlem2 20587* Lemma for dchrisum 20589. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisumlem3 20588* Lemma for dchrisum 20589. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-May-2016.)
ℤ/n       RHom              DChr                                                                                    ..^ ..^

Theoremdchrisum 20589* 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 20590* Lemma for dchrmusum 20621 and dchrisumn0 20618. Apply dchrisum 20589 for the function . (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrmusum2 20591* 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 20592* 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 20593* Give an expression for remarkably similar to Λ given in dchrvmasumlem1 20592. Part of Lemma 9.4.3 of [Shapiro], p. 380. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasum2if 20594* Combine the results of dchrvmasumlem1 20592 and dchrvmasum2lem 20593 inside a conditional. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr                                                 Λ

Theoremdchrvmasumlem2 20595* Lemma for dchrvmasum 20622. (Contributed by Mario Carneiro, 4-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlem3 20596* Lemma for dchrvmasum 20622. (Contributed by Mario Carneiro, 3-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumlema 20597* Lemma for dchrvmasum 20622 and dchrvmasumif 20600. Apply dchrisum 20589 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 20598* Lemma for dchrvmasumif 20600. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr

Theoremdchrvmasumiflem2 20599* Lemma for dchrvmasum 20622. (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                                                           Λ

Theoremdchrvmasumif 20600* 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 20622.) (Contributed by Mario Carneiro, 5-May-2016.)
ℤ/n       RHom              DChr                                                               Λ

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