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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremppiublem2 23201 A prime greater than does not divide or , so its residue is or . (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremppiub 23202 An upper bound on the prime-counting function π, which counts the number of primes less than . (Contributed by Mario Carneiro, 13-Mar-2014.)
π

Theoremvmalelog 23203 The von Mangoldt function is less than the natural log. (Contributed by Mario Carneiro, 7-Apr-2016.)
Λ

Theoremchtlepsi 23204 The first Chebyshev function is less than the second. (Contributed by Mario Carneiro, 7-Apr-2016.)
ψ

Theoremchprpcl 23205 Closure of the second Chebyshev function in the positive reals. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremchpeq0 23206 The second Chebyshev function is zero iff its argument is less than . (Contributed by Mario Carneiro, 9-Apr-2016.)
ψ

Theoremchteq0 23207 The first Chebyshev function is zero iff its argument is less than . (Contributed by Mario Carneiro, 9-Apr-2016.)

Theoremchtleppi 23208 Upper bound on the function. (Contributed by Mario Carneiro, 22-Sep-2014.)
π

Theoremchtublem 23209 Lemma for chtub 23210. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theoremchtub 23210 An upper bound on the Chebyshev function. (Contributed by Mario Carneiro, 13-Mar-2014.) (Revised 22-Sep-2014.)

Theoremfsumvma 23211* Rewrite a sum over the von Mangoldt function as a sum over prime powers. (Contributed by Mario Carneiro, 15-Apr-2016.)
Λ

Theoremfsumvma2 23212* Apply fsumvma 23211 for the common case of all numbers less than a real number . (Contributed by Mario Carneiro, 30-Apr-2016.)
Λ

Theorempclogsum 23213* The logarithmic analogue of pcprod 14264. 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 23214* 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 23215* 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 23216* 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 23217* 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 23218 An upper bound on the second Chebyshev function. (Contributed by Mario Carneiro, 8-Apr-2016.)
ψ

Theoremlogfacubnd 23219 A simple upper bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfaclbnd 23220 A lower bound on the logarithm of a factorial. (Contributed by Mario Carneiro, 16-Apr-2016.)

Theoremlogfacbnd3 23221 Show the stronger statement alluded to in logfacrlim 23222. (Contributed by Mario Carneiro, 20-May-2016.)

Theoremlogfacrlim 23222 Combine the estimates logfacubnd 23219 and logfaclbnd 23220, 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 23223* The sum has the asymptotic expansion . (More precisely, the omitted term has order .) (Contributed by Mario Carneiro, 22-May-2016.)

Theoremlogfacrlim2 23224* Write out logfacrlim 23222 as a sum of logs. (Contributed by Mario Carneiro, 18-May-2016.) (Revised by Mario Carneiro, 22-May-2016.)

14.4.5  Perfect Number Theorem

Theoremmersenne 23225 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 23226 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 23227 Lemma for perfect 23229. (Contributed by Mario Carneiro, 7-Jun-2016.)

Theoremperfectlem2 23228 Lemma for perfect 23229. (Contributed by Mario Carneiro, 17-May-2016.)

Theoremperfect 23229* 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.) This is Metamath 100 proof #70. (Contributed by Mario Carneiro, 17-May-2016.)

14.4.6  Characters of Z/nZ

Syntaxcdchr 23230 Extend class notation with the group of Dirichlet characters.
DChr

Definitiondf-dchr 23231* The group of Dirichlet characters is the set of monoid homomorphisms from to the multiplicative monoid of the complex numbers, equipped with the group operation of pointwise multiplication. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr ℤ/n mulGrp MndHom mulGrpfld Unit

Theoremdchrval 23232* Value of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              Unit              mulGrp MndHom mulGrpfld

Theoremdchrbas 23233* Base set of the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n              Unit                     mulGrp MndHom mulGrpfld

Theoremdchrelbas 23234 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 23235* 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 23236* 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 23237* 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 23238 Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.)
DChr

Theoremdchrmhm 23239 A Dirichlet character is a monoid homomorphism. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr       ℤ/n              mulGrp MndHom mulGrpfld

Theoremdchrf 23240 A Dirichlet character is a function. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrelbas4 23241* 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 23242 Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.)
DChr       ℤ/n              RHom

Theoremdchrzrhcl 23243 A Dirichlet character takes values in the complex numbers. (Contributed by Mario Carneiro, 12-May-2016.)
DChr       ℤ/n              RHom

Theoremdchrzrhmul 23244 A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.)
DChr       ℤ/n              RHom

Theoremdchrplusg 23245 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrmul 23246 Group operation on the group of Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrmulcl 23247 Closure of the group operation on Dirichlet characters. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n

Theoremdchrn0 23248 A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchr1cl 23249* Closure of the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchrmulid2 23250* Left identity for the principal Dirichlet character. (Contributed by Mario Carneiro, 18-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchrinvcl 23251* Closure of the group inverse operation on Dirichlet characters. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr       ℤ/n                     Unit

Theoremdchrabl 23252 The set of Dirichlet characters is an Abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr

Theoremdchrfi 23253 The group of Dirichlet characters is a finite group. (Contributed by Mario Carneiro, 19-Apr-2016.)
DChr

Theoremdchrghm 23254 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 23255 Value of the principal Dirichlet character. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchreq 23256* A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchrresb 23257 A Dirichlet character is determined by its values on the unit group. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n              Unit

Theoremdchrabs 23258 A Dirichlet character takes values on the unit circle. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr                     ℤ/n       Unit

Theoremdchrinv 23259 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 23260 A Dirichlet character takes values inside the unit circle. (Contributed by Mario Carneiro, 3-May-2016.)
DChr              ℤ/n

Theoremdchr1re 23261 The principal Dirichlet character is a real character. (Contributed by Mario Carneiro, 2-May-2016.)
DChr       ℤ/n

Theoremdchrptlem1 23262* Lemma for dchrpt 23265. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                                          Unit       mulGrps        .g                     Word        DProd        DProd        dProj

Theoremdchrptlem2 23263* Lemma for dchrpt 23265. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                                          Unit       mulGrps        .g                     Word        DProd        DProd        dProj

Theoremdchrptlem3 23264* Lemma for dchrpt 23265. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr       ℤ/n                                          Unit       mulGrps        .g                     Word        DProd        DProd

Theoremdchrpt 23265* 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 23266* 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 23267* 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 23268* Lemma for sumdchr 23270. (Contributed by Mario Carneiro, 28-Apr-2016.)
DChr              ℤ/n

Theoremdchrhash 23269 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 23270* 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 23271* 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 23272* 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

14.4.7  Bertrand's postulate

Theorembcctr 23273 Value of the central binomial coefficient. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcbcctr 23274* Prime count of a central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembcmono 23275 The binomial coefficient is monotone in its second argument, up to the midway point. (Contributed by Mario Carneiro, 5-Mar-2014.)

Theorembcmax 23276 The binomial coefficient takes its maximum value at the center. (Contributed by Mario Carneiro, 5-Mar-2014.)

Theorembcp1ctr 23277 Ratio of two central binomial coefficients. (Contributed by Mario Carneiro, 10-Mar-2014.)

Theorembclbnd 23278 A bound on the binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theoremefexple 23279 Convert a bound on a power to a bound on the exponent. (Contributed by Mario Carneiro, 11-Mar-2014.)

Theorembpos1lem 23280* Lemma for bpos1 23281. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembpos1 23281* 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 23282 An upper bound on the prime powers dividing a central binomial coefficient. (Contributed by Mario Carneiro, 9-Mar-2014.)

Theorembposlem2 23283 There are no odd primes in the range dividing the -th central binomial coefficient. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorembposlem3 23284* Lemma for bpos 23291. Since the binomial coefficient does not have any primes in the range or by bposlem2 23283 and prmfac1 14109, 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 23285* Lemma for bpos 23291. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem5 23286* Lemma for bpos 23291. Bound the product of all small primes in the binomial coefficient. (Contributed by Mario Carneiro, 15-Mar-2014.)

Theorembposlem6 23287* Lemma for bpos 23291. 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 23288* Lemma for bpos 23291. The function is decreasing. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorembposlem8 23289 Lemma for bpos 23291. Evaluate and show it is less than . (Contributed by Mario Carneiro, 14-Mar-2014.)
; ;

Theorembposlem9 23290* Lemma for bpos 23291. Derive a contradiction. (Contributed by Mario Carneiro, 14-Mar-2014.)
;

Theorembpos 23291* 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. This is Metamath 100 proof #98. (Contributed by Mario Carneiro, 14-Mar-2014.)

14.4.8  Legendre symbol

Syntaxclgs 23292 Extend class notation with the Legendre symbol function.

Definitiondf-lgs 23293* Define the Legendre symbol (actually the Kronecker symbol, which extends the Legendre symbol to all integers). (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem1 23294 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 23295 The set of all integers with absolute value at most contains . (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem3 23296* The set of all integers with absolute value at most is closed under multiplication. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgslem4 23297* 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 23298* Value of the Legendre symbol at an arbitrary integer. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfval 23299* Value of the function which defines the Legendre symbol at the primes. (Contributed by Mario Carneiro, 4-Feb-2015.)

Theoremlgsfcl2 23300* 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.)

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800
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