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

Theorempclogsum 21001* The logarithmic analogue of pcprod 13266. The sum of the logarithms of the primes dividing multiplied by their powers yields the logarithm of . (Contributed by Mario Carneiro, 15-Apr-2016.)

Theoremvmasum 21002* The sum of the von Mangoldt function over the divisors of . Equation 9.2.4 of [Shapiro], p. 328. (Contributed by Mario Carneiro, 15-Apr-2016.)
Λ

Theoremlogfac2 21003* Another expression for the logarithm of a factorial, in terms of the von Mangoldt function. Equation 9.2.7 of [Shapiro], p. 329. (Contributed by Mario Carneiro, 15-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
Λ

Theoremchpval2 21004* Express the second Chebyshev function directly as a sum over the primes less than (instead of indirectly through the von Mangoldt function). (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpchtsum 21005* The second Chebyshev function is the sum of the theta function at arguments quickly approaching zero. (This is usually stated as an infinite sum, but after a certain point, the terms are all zero, and it is easier for us to use an explicit finite sum.) (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchpub 21006 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremlogfacubnd 21007 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfaclbnd 21008 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfacbnd3 21009 Show the stronger statement alluded to in logfacrlim 21010. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremlogfacrlim 21010 Combine the estimates logfacubnd 21007 and logfaclbnd 21008, to get . Equation 9.2.9 of [Shapiro], p. 329. This is a weak form of the even stronger statement, . (Contributed by Mario Carneiro, 16-Apr-2016.) (Revised by Mario Carneiro, 21-May-2016.)

Theoremlogexprlim 21011* The sum has the asymptotic expansion . (More precisely, the omitted term has order .) (Contributed by Mario Carneiro, 22-May-2016.)

Theoremlogfacrlim2 21012* Write out logfacrlim 21010 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.)

13.4.5  Perfect Number Theorem

Theoremmersenne 21013 A Mersenne prime is a prime number of the form . This theorem shows that the in this expression is necessarily also prime. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremperfect1 21014 Euclid's contribution to the Euclid-Euler theorem. A number of the form is a perfect number. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremperfectlem1 21015 Lemma for perfect 21017. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremperfectlem2 21016 Lemma for perfect 21017. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremperfect 21017* The Euclid-Euler theorem, or Perfect Number theorem. A positive even integer is a perfect number (that is, its divisor sum is ) if and only if it is of the form , where is prime (a Mersenne prime). (It follows from this that is also prime.) (Contributed by Mario Carneiro, 17-May-2016.)

13.4.6  Characters of Z/nZ

Syntaxcdchr 21018 Extend class notation with the group of Dirichlet characters.
DChr

Definitiondf-dchr 21019* The group of Dirichlet characters is the set of monoid homomorphisms from to the multiplicative monoid of the complexes, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr ℤ/n mulGrp MndHom mulGrpfld Unit

Theoremdchrval 21020* Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              Unit              mulGrp MndHom mulGrpfld

Theoremdchrbas 21021* Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              Unit                     mulGrp MndHom mulGrpfld

Theoremdchrelbas 21022 A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of , which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              Unit                     mulGrp MndHom mulGrpfld

Theoremdchrelbas2 21023* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of , which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              Unit                     mulGrp MndHom mulGrpfld

Theoremdchrelbas3 21024* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of , which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchrelbasd 21025* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of , which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchrrcl 21026 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
DChr

Theoremdchrmhm 21027 A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr       ℤ/n              mulGrp MndHom mulGrpfld

Theoremdchrf 21028 A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrelbas4 21029* A Dirichlet character is a monoid homomorphism from the multiplicative monoid on ℤ/nℤ to the multiplicative monoid of , which is zero off the group of units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              RHom       mulGrp MndHom mulGrpfld

Theoremdchrzrh1 21030 Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)
DChr       ℤ/n              RHom

Theoremdchrzrhcl 21031 A Dirichlet character takes values in the complexes. (Contributed by Mario Carneiro, 12-May-2016.)
DChr       ℤ/n              RHom

Theoremdchrzrhmul 21032 A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)
DChr       ℤ/n              RHom

Theoremdchrplusg 21033 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrmul 21034 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrmulcl 21035 Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrn0 21036 A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchr1cl 21037* Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchrmulid2 21038* Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchrinvcl 21039* Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchrabl 21040 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr

Theoremdchrfi 21041 The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr

Theoremdchrghm 21042 A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.)
DChr       ℤ/n              Unit       mulGrps        mulGrpflds

Theoremdchr1 21043 Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchreq 21044* A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchrresb 21045 A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchrabs 21046 A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr                     ℤ/n       Unit

Theoremdchrinv 21047 The inverse of a Dirichlet character is the conjugate (which is also the multiplicative inverse, because the values of are unimodular). (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr

Theoremdchrabs2 21048 A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.)
DChr              ℤ/n

Theoremdchr1re 21049 The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.)
DChr       ℤ/n

Theoremdchrptlem1 21050* Lemma for dchrpt 21053. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                                          Unit       mulGrps        .g                     Word        DProd        DProd        dProj

Theoremdchrptlem2 21051* Lemma for dchrpt 21053. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                                          Unit       mulGrps        .g                     Word        DProd        DProd        dProj

Theoremdchrptlem3 21052* Lemma for dchrpt 21053. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                                          Unit       mulGrps        .g                     Word        DProd        DProd

Theoremdchrpt 21053* For any element other than 1, there is a Dirichlet character that is not one at the given element. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n

Theoremdchrsum2 21054* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character is if is non-principal and otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                            Unit

Theoremdchrsum 21055* An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character is if is non-principal and otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n

Theoremsumdchr2 21056* Lemma for sumdchr 21058. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr              ℤ/n

Theoremdchrhash 21057 There are exactly Dirichlet characters modulo . Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr

Theoremsumdchr 21058* An orthogonality relation for Dirichlet characters: the sum of for fixed and all is if and otherwise. Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr              ℤ/n

Theoremdchr2sum 21059* An orthogonality relation for Dirichlet characters: the sum of over all is nonzero only when . Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n

Theoremsum2dchr 21060* An orthogonality relation for Dirichlet characters: the sum of for fixed and all is if and otherwise. Part of Theorem 6.5.2 of [Shapiro] p. 232. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr              ℤ/n              Unit

13.4.7  Bertrand's postulate

Theorembcctr 21061 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcbcctr 21062* Prime count of a central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembcmono 21063 The binomial coefficient is monotone in its second argument, up to the midway point. (Contributed by Mario Carneiro, 5-Mar-2014.)

Theorembcmax 21064 The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.)

Theorembcp1ctr 21065 Ratio of two central binomial coefficients. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembclbnd 21066 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremefexple 21067 Convert a bound on a power to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theorembpos1lem 21068* Lemma for bpos1 21069. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembpos1 21069* 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 21070 An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.)

Theorembposlem2 21071 There are no odd primes in the range dividing the -th central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembposlem3 21072* Lemma for bpos 21079. Since the binomial coefficient does not have any primes in the range or by bposlem2 21071 and prmfac1 13120, 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 21073* Lemma for bpos 21079. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem5 21074* Lemma for bpos 21079. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembposlem6 21075* Lemma for bpos 21079. 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 21076* Lemma for bpos 21079. The function is decreasing. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem8 21077 Lemma for bpos 21079. Evaluate and show it is less than . (Contributed by Mario Carneiro, 14-Mar-2014.)
; ;

Theorembposlem9 21078* Lemma for bpos 21079. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.)
;

Theorembpos 21079* 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 21080 Extend class notation with the Legendre symbol function.

Definitiondf-lgs 21081* Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem1 21082 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 21083 The set of all integers with absolute value at most contains . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem3 21084* The set of all integers with absolute value at most is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem4 21085* 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 21086* Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfval 21087* Value of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfcl2 21088* 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 21089* The Legendre symbol is an element of . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfcl 21090* Closure of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfle1 21091* The function has magnitude less or equal to . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval2lem 21092* Lemma for lgsval2 21098. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval4lem 21093* Lemma for lgsval4 21102. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgscl2 21094* The Legendre symbol is an integer with absolute value less or equal to . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgs0 21095 The Legendre symbol when the second argument is zero. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgscl 21096 The Legendre symbol is an integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsle1 21097 The Legendre symbol has absolute value less or equal to . Together with lgscl 21096 this implies that it takes values in . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval2 21098 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 21099 The Legendre symbol at . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsval3 21100 The Legendre symbol at an odd prime (this is the traditional domain of the Legendre symbol). (Contributed by Mario Carneiro, 4-Feb-2015.)

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