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Theorem List for Metamath Proof Explorer - 13201-13300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempcval 13201* The value of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempceulem 13202* Lemma for pceu 13203. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceu 13203* Uniqueness for the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczpre 13204* Connect the prime count pre-function to the actual prime count function, when restricted to the integers. (Contributed by Mario Carneiro, 23-Feb-2014.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)

Theorempczcl 13205 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccl 13206 Closure of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempccld 13207 Closure of the prime power function. (Contributed by Mario Carneiro, 29-May-2016.)

Theorempcmul 13208 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdiv 13209 Division property of the prime power function. (Contributed by Mario Carneiro, 1-Mar-2014.)

Theorempcqmul 13210 Multiplication property of the prime power function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempc0 13211 The value of the prime power function at zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc1 13212 Value of the prime count function at 1. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqcl 13213 Closure of the general prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcqdiv 13214 Division property of the prime power function. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcrec 13215 Prime power of a reciprocal. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcexp 13216 Prime power of an exponential. (Contributed by Mario Carneiro, 10-Aug-2015.)

Theorempcxcl 13217 Extended real closure of the general prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcge0 13218 The prime count of an integer is greater or equal to zero. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempczdvds 13219 Defining property of the prime count function. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcdvds 13220 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds 13221 Defining property of the prime count function. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcndvds 13222 Defining property of the prime count function. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempczndvds2 13223 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcndvds2 13224 The remainder after dividing out all factors of is not divisible by . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcdvdsb 13225 divides if and only if is at most the count of . (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcelnn 13226 There are a positive number of powers of a prime in iff divides . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempceq0 13227 There are zero powers of a prime in iff does not divide . (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcidlem 13228 The prime count of a prime power. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcid 13229 The prime count of a prime power. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcneg 13230 The prime count of a negative number. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcabs 13231 The prime count of an absolute value. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcdvdstr 13232 The prime count increases under the divisibility relation. (Contributed by Mario Carneiro, 13-Mar-2014.)

Theorempcgcd1 13233 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempcgcd 13234 The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theorempc2dvds 13235* A characterization of divisibility in terms of prime count. (Contributed by Mario Carneiro, 23-Feb-2014.) (Revised by Mario Carneiro, 3-Oct-2014.)

Theorempc11 13236* The prime count function, viewed as a function from to , is one-to-one. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcz 13237* The prime count function can be used as an indicator that a given rational number is an integer. (Contributed by Mario Carneiro, 23-Feb-2014.)

Theorempcprmpw2 13238* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempcprmpw 13239* Self-referential expression for a prime power. (Contributed by Mario Carneiro, 16-Jan-2015.)

Theorempcaddlem 13240 Lemma for pcadd 13241. The original numbers and have been decomposed using the prime count function as where are both not divisible by and , and similarly for . (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcadd 13241 An inequality for the prime count of a sum. This is the source of the ultrametric inequality for the p-adic metric. (Contributed by Mario Carneiro, 9-Sep-2014.)

Theorempcadd2 13242 The inequality of pcadd 13241 becomes an equality when one of the factors has prime count strictly less than the other. (Contributed by Mario Carneiro, 16-Jan-2015.) (Revised by Mario Carneiro, 26-Jun-2015.)

Theorempcmptcl 13243 Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcmpt 13244* Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcmpt2 13245* Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theorempcmptdvds 13246 The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcprod 13247* The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremsumhash 13248* The sum of 1 over a set is the size of the set. (Contributed by Mario Carneiro, 8-Mar-2014.) (Revised by Mario Carneiro, 20-May-2014.)

Theoremfldivp1 13249 The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)

Theorempcfaclem 13250 Lemma for pcfac 13251. (Contributed by Mario Carneiro, 20-May-2014.)

Theorempcfac 13251* Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Theorempcbc 13252* Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Theoremqexpz 13253 If a power of a rational number is an integer, then the number is an integer. In other words, all n-th roots are irrational unless they are integers (so that the original number is an n-th power). (Contributed by Mario Carneiro, 10-Aug-2015.)

Theoremexpnprm 13254 A second or higher power of a rational number is not a prime number. Or by contraposition, the n-th root of a prime number is irrational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)

6.2.6  Pocklington's theorem

Theoremprmpwdvds 13255 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthlem 13256 Lemma for pockthg 13257. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthg 13257* The generalized Pocklington's theorem. If where , then is prime if and only if for every prime factor of , there is an such that and . (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthi 13258 Pocklington's theorem, which gives a sufficient criterion for a number to be prime. This is the preferred method for verifying large primes, being much more efficient to compute than trial division. This form has been optimized for application to specific large primes; see pockthg 13257 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)

6.2.7  Infinite primes theorem

Theoremunbenlem 13259* Lemma for unben 13260. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremunben 13260* An unbounded set of natural numbers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoreminfpnlem1 13261* Lemma for infpn 13263. The smallest divisor (greater than 1) of is a prime greater than . (Contributed by NM, 5-May-2005.)

Theoreminfpnlem2 13262* Lemma for infpn 13263. For any natural number , there exists a prime number greater than . (Contributed by NM, 5-May-2005.)

Theoreminfpn 13263* There exist infinitely many prime numbers: for any natural number , there exists a prime number greater than . (See infpn2 13264 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)

Theoreminfpn2 13264* There exist infinitely many prime numbers: the set of all primes is unbounded by infpn 13263, so by unben 13260 it is infinite. (Contributed by NM, 5-May-2005.)

Theoremprmunb 13265* The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremprminf 13266 There are an infinite number of primes. (Contributed by Paul Chapman, 28-Nov-2012.)

6.2.8  Sum of prime reciprocals

Theoremprmreclem1 13267* Lemma for prmrec 13273. Properties of the "square part" function, which extracts the of the decomposition , with maximal and squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremprmreclem2 13268* Lemma for prmrec 13273. There are at most squarefree numbers which divide no primes larger than . (We could strengthen this to but there's no reason to.) We establish the inequality by showing that the prime counts of the number up to completely determine it because all higher prime counts are zero, and they are all at most because no square divides the number, so there are at most possibilities. (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremprmreclem3 13269* Lemma for prmrec 13273. The main inequality established here is , where is the set of squarefree numbers in . This is demonstrated by the map where is the largest number whose square divides . (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremprmreclem4 13270* Lemma for prmrec 13273. Show by induction that the indexed (nondisjoint) union is at most the size of the prime reciprocal series. The key counting lemma is hashdvds 13147, to show that the number of numbers in that divide is at most . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmreclem5 13271* Lemma for prmrec 13273. Here we show the inequality by decomposing the set into the disjoint union of the set of those numbers that are not divisible by any "large" primes (above ) and the indexed union over of the numbers that divide the prime . By prmreclem4 13270 the second of these has size less than times the prime reciprocal series, which is less than by assumption, we find that the complementary part must be at least large. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmreclem6 13272* Lemma for prmrec 13273. If the series was convergent, there would be some such that the sum starting from sums to less than ; this is a sufficient hypothesis for prmreclem5 13271 to produce the contradictory bound , which is false for . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmrec 13273* The sum of the reciprocals of the primes diverges. This is the "second" proof at http://en.wikipedia.org/wiki/Prime_harmonic_series, attributed to Paul Erdős. (Contributed by Mario Carneiro, 6-Aug-2014.)

6.2.9  Fundamental theorem of arithmetic

Theorem1arithlem1 13274* Lemma for 1arith 13278. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem2 13275* Lemma for 1arith 13278. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem3 13276* Lemma for 1arith 13278. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem4 13277* Lemma for 1arith 13278. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arith 13278* Fundamental theorem of arithmetic, where a prime factorization is represented as a sequence of prime exponents, for which only finitely many primes have nonzero exponent. The function maps the set of positive integers one-to-one onto the set of prime factorizations . (Contributed by Paul Chapman, 17-Nov-2012.) (Proof shortened by Mario Carneiro, 30-May-2014.)

Theorem1arith2 13279* Fundamental theorem of arithmetic, where a prime factorization is represented as a finite monotonic 1-based sequence of primes. Every positive integer has a unique prime factorization. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.)

6.2.10  Lagrange's four-square theorem

Syntaxcgz 13280 Extend class notation with the set of gaussian integers.

Definitiondf-gz 13281 Define the set of gaussian integers, which are complex numbers whose real and imaginary parts are integers. (Note that the is actually part of the symbol token and has no independent meaning.) (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremelgz 13282 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcn 13283 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremzgz 13284 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremigz 13285 is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgznegcl 13286 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcjcl 13287 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzaddcl 13288 The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzmulcl 13289 The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzreim 13290 Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theoremgzsubcl 13291 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzabssqcl 13292 The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem5 13293 Lemma for 4sq 13315. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem6 13294 Lemma for 4sq 13315. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem7 13295 Lemma for 4sq 13315. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem8 13296 Lemma for 4sq 13315. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem9 13297 Lemma for 4sq 13315. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem10 13298 Lemma for 4sq 13315. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem1 13299* Lemma for 4sq 13315. The set is the set of all numbers that are expressible as a sum of four squares. Our goal is to show that ; here we show one subset direction. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem2 13300* Lemma for 4sq 13315. Change bound variables in . (Contributed by Mario Carneiro, 14-Jul-2014.)

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