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Theorem List for Metamath Proof Explorer - 21001-21100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembcmax 21001 The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.)

Theorembcp1ctr 21002 Ratio of two central binomial coefficients. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembclbnd 21003 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremefexple 21004 Convert a bound on a power to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theorembpos1lem 21005* Lemma for bpos1 21006. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembpos1 21006* Bertrand's postulate, checked numerically for , using the prime sequence . (Contributed by Mario Carneiro, 12-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
;

Theorembposlem1 21007 An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.)

Theorembposlem2 21008 There are no odd primes in the range dividing the -th central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembposlem3 21009* Lemma for bpos 21016. Since the binomial coefficient does not have any primes in the range or by bposlem2 21008 and prmfac1 13059, respectively, and it does not have any in the range by hypothesis, the product of the primes up through must be sufficient to compose the whole binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem4 21010* Lemma for bpos 21016. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem5 21011* Lemma for bpos 21016. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembposlem6 21012* Lemma for bpos 21016. By using the various bounds at our disposal, arrive at an inequality that is false for large enough. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theorembposlem7 21013* Lemma for bpos 21016. The function is decreasing. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem8 21014 Lemma for bpos 21016. Evaluate and show it is less than . (Contributed by Mario Carneiro, 14-Mar-2014.)
; ;

Theorembposlem9 21015* Lemma for bpos 21016. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.)
;

Theorembpos 21016* Bertrand's postulate: there is a prime between and for every positive integer . This proof follows Erdős's method, for the most part, but with some refinements due to Shigenori Tochiori to save us some calculations of large primes. See http://en.wikipedia.org/wiki/Proof_of_Bertrand%27s_postulate for an overview of the proof strategy. (Contributed by Mario Carneiro, 14-Mar-2014.)

13.4.8  Legendre symbol

Syntaxclgs 21017 Extend class notation with the Legendre symbol function.

Definitiondf-lgs 21018* Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem1 21019 When is coprime to the prime , is equivalent to or , and so adding makes it equivalent to or . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem2 21020 The set of all integers with absolute value at most contains . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem3 21021* The set of all integers with absolute value at most is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem4 21022* The function is closed in integers with absolute value less than (namely although this representation is less useful to us). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval 21023* Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfval 21024* Value of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfcl2 21025* The function is closed in integers with absolute value less than (namely although this representation is less useful to us). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgscllem 21026* The Legendre symbol is an element of . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfcl 21027* Closure of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfle1 21028* The function has magnitude less or equal to . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval2lem 21029* Lemma for lgsval2 21035. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4lem 21030* Lemma for lgsval4 21039. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgscl2 21031* The Legendre symbol is an integer with absolute value less or equal to . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgs0 21032 The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgscl 21033 The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsle1 21034 The Legendre symbol has absolute value less or equal to . Together with lgscl 21033 this implies that it takes values in . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval2 21035 The Legendre symbol at a prime (this is the traditional domain of the Legendre symbol, except for the addition of prime ). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgs2 21036 The Legendre symbol at . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval3 21037 The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsvalmod 21038 The Legendre symbol is equivalent to , . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4 21039* Restate lgsval 21023 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 21040* Closure of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4a 21041* Same as lgsval4 21039 for positive . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsneg 21042 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 21043 The Legendre symbol for nonnegative first parameter is unchanged by negation of the second. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsmod 21044 The Legendre (Jacobi) symbol is preserved under reduction when is odd. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdilem 21045 Lemma for lgsdi 21055 and lgsdir 21053: the sign part of the Legendre symbol is multiplicative. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem1 21046 Lemma for lgsdir2 21051. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem2 21047 Lemma for lgsdir2 21051. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem3 21048 Lemma for lgsdir2 21051. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem4 21049 Lemma for lgsdir2 21051. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2lem5 21050 Lemma for lgsdir2 21051. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir2 21051 The Legendre symbol is completely multiplicative at . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdirprm 21052 The Legendre symbol is completely multiplicative at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdir 21053 The Legendre symbol is completely multiplicative in its left argument. Together with lgsqr 21069 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 21054* Lemma for lgsdi 21055. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsdi 21055 The Legendre symbol is completely multiplicative in its right argument. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgsne0 21056 The Legendre symbol is nonzero (and hence equal to or ) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgsabs1 21057 The Legendre symbol is nonzero (and hence equal to or ) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgssq 21058 The Legendre symbol at a square is equal to . Together with lgsmod 21044 this implies that the Legendre symbol takes value at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.)

Theoremlgssq2 21059 The Legendre symbol at a square is equal to . (Contributed by Mario Carneiro, 5-Feb-2015.)

Theorem1lgs 21060 The Legendre symbol at . (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgs1 21061 The Legendre symbol at . (Contributed by Mario Carneiro, 28-Apr-2016.)

Theoremlgsdirnn0 21062 Variation on lgsdir 21053 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 21063 Variation on lgsdi 21055 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 21064 Lemma for lgsqr 21069. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem2 21065* Lemma for lgsqr 21069. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem3 21066* Lemma for lgsqr 21069. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem4 21067* Lemma for lgsqr 21069. (Contributed by Mario Carneiro, 15-Jun-2015.)
ℤ/n       Poly1              deg1        eval1       .gmulGrp       var1                            RHom

Theoremlgsqrlem5 21068* Lemma for lgsqr 21069. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsqr 21069* The Legendre symbol for odd primes is iff the number is not a multiple of the prime (in which case it is , see lgsne0 21056) and the number is a quadratic residue (it is for nonresidues by the process of elimination from lgsabs1 21057). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)

Theoremlgsdchrval 21070* 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 21071* 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 21072* Lemma for lgseisen 21076. 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 21073* Lemma for lgseisen 21076. The function is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)

Theoremlgseisenlem3 21074* Lemma for lgseisen 21076. (Contributed by Mario Carneiro, 17-Jun-2015.)
ℤ/n       mulGrp       RHom       g

Theoremlgseisenlem4 21075* Lemma for lgseisen 21076. 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 21076* Eisenstein's lemma, an expression for when are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquadlem1 21077* Lemma for lgsquad 21080. Count the members of with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquadlem2 21078* Lemma for lgsquad 21080. Count the members of with even coordinates, and combine with lgsquadlem1 21077 to get the total count of lattice points in (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremlgsquad 21080 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 21083 Extend lgsquad 21080 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremlgsquad3 21084 Extend lgsquad2 21083 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremm1lgs 21085 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 21086* Lemma for 2sq 21099. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem2 21087* Lemma for 2sq 21099. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theoremmul2sq 21088 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 21089 Lemma for 2sqlem5 21091. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem4 21090 Lemma for 2sqlem5 21091. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem5 21091 Lemma for 2sq 21099. 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 21092* Lemma for 2sq 21099. 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 21093* Lemma for 2sq 21099. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem8a 21094* Lemma for 2sqlem8 21095. (Contributed by Mario Carneiro, 4-Jun-2016.)

Theorem2sqlem8 21095* Lemma for 2sq 21099. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqlem9 21096* Lemma for 2sq 21099. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sqlem10 21097* Lemma for 2sq 21099. 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 21098* Lemma for 2sq 21099. (Contributed by Mario Carneiro, 19-Jun-2015.)

Theorem2sq 21099* All primes of the form are sums of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)

Theorem2sqblem 21100 The converse to 2sq 21099. (Contributed by Mario Carneiro, 20-Jun-2015.)

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