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Theorem List for Metamath Proof Explorer - 12901-13000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempcadd2 12901 The inequality of pcadd 12900 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 12902 Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcmpt 12903* Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcmpt2 12904* Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)

Theorempcmptdvds 12905 The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theorempcprod 12906* The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)

Theoremsumhash 12907* 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 12908 The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)

Theorempcfaclem 12909 Lemma for pcfac 12910. (Contributed by Mario Carneiro, 20-May-2014.)

Theorempcfac 12910* Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Theorempcbc 12911* Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)

Theoremqexpz 12912 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 12913 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 12914 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthlem 12915 Lemma for pockthg 12916. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthg 12916* 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 12917 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 12916 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)

6.2.7  Infinite primes theorem

Theoremunbenlem 12918* Lemma for unben 12919. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremunben 12919* An unbounded set of natural numbers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoreminfpnlem1 12920* Lemma for infpn 12922. The smallest divisor (greater than 1) of is a prime greater than . (Contributed by NM, 5-May-2005.)

Theoreminfpnlem2 12921* Lemma for infpn 12922. For any natural number , there exists a prime number greater than . (Contributed by NM, 5-May-2005.)

Theoreminfpn 12922* There exist infinitely many prime numbers: for any natural number , there exists a prime number greater than . (See infpn2 12923 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)

Theoreminfpn2 12923* There exist infinitely many prime numbers: the set of all primes is unbounded by infpn 12922, so by unben 12919 it is infinite. (Contributed by NM, 5-May-2005.)

Theoremprmunb 12924* The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremprminf 12925 There are an infinite number of primes. (Contributed by Paul Chapman, 28-Nov-2012.)

6.2.8  Sum of prime reciprocals

Theoremprmreclem1 12926* Lemma for prmrec 12932. Properties of the "square part" function, which extracts the of the decomposition , with maximal and squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremprmreclem2 12927* Lemma for prmrec 12932. 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 12928* Lemma for prmrec 12932. 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 12929* Lemma for prmrec 12932. Show by induction that the indexed (nondisjoint) union is at most the size of the prime reciprocal series. The key counting lemma is hashdvds 12806, to show that the number of numbers in that divide is at most . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmreclem5 12930* Lemma for prmrec 12932. 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 12929 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 12931* Lemma for prmrec 12932. 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 12930 to produce the contradictory bound , which is false for . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmrec 12932* 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 12933* Lemma for 1arith 12937. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem2 12934* Lemma for 1arith 12937. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem3 12935* Lemma for 1arith 12937. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem4 12936* Lemma for 1arith 12937. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arith 12937* 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 12938* 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 12939 Extend class notation with the set of gaussian integers.

Definitiondf-gz 12940 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 12941 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcn 12942 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremzgz 12943 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremigz 12944 is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgznegcl 12945 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcjcl 12946 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzaddcl 12947 The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzmulcl 12948 The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzreim 12949 Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theoremgzsubcl 12950 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzabssqcl 12951 The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem5 12952 Lemma for 4sq 12974. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem6 12953 Lemma for 4sq 12974. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem7 12954 Lemma for 4sq 12974. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem8 12955 Lemma for 4sq 12974. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem9 12956 Lemma for 4sq 12974. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem10 12957 Lemma for 4sq 12974. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem1 12958* Lemma for 4sq 12974. 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 12959* Lemma for 4sq 12974. Change bound variables in . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem3 12960* Lemma for 4sq 12974. Sufficient condition to be in . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem4a 12961* Lemma for 4sqlem4 12962. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem4 12962* Lemma for 4sq 12974. We can express the four-square property more compactly in terms of gaussian integers, because the norms of gaussian integers are exactly sums of two squares. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremmul4sqlem 12963* Lemma for mul4sq 12964: algebraic manipulations. The extra assumptions involving are for a part of 4sqlem17 12971 which needs to know not just that the product is a sum of squares, but also that it preserves divisibility by . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremmul4sq 12964* Euler's four-square identity: The product of two sums of four squares is also a sum of four squares. This is usually quoted as an explicit formula involving eight real variables; we save some time by working with complex numbers (gaussian integers) instead, so that we only have to work with four variables, and also hiding the actual formula for the product in the proof of mul4sqlem 12963. (For the curious, the explicit formula that is used is .) (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem11 12965* Lemma for 4sq 12974. Use the pigeonhole principle to show that the sets and have a common element, . (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem12 12966* Lemma for 4sq 12974. For any odd prime , there is a such that is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem13 12967* Lemma for 4sq 12974. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem14 12968* Lemma for 4sq 12974. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem15 12969* Lemma for 4sq 12974. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem16 12970* Lemma for 4sq 12974. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem17 12971* Lemma for 4sq 12974. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem18 12972* Lemma for 4sq 12974. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem19 12973* Lemma for 4sq 12974. The proof is by strong induction - we show that if all the integers less than are in , then is as well. In this part of the proof we do the induction argument and dispense with all the cases except the odd prime case, which is sent to 4sqlem18 12972. If is , we show directly; otherwise if is composite, is the product of two numbers less than it (and hence in by assumption), so by mul4sq 12964 . (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theorem4sq 12974* Lagrange's four-square theorem, or Bachet's conjecture: every nonnegative integer is expressible as a sum of four squares. (Contributed by Mario Carneiro, 16-Jul-2014.)

6.2.11  Van der Waerden's theorem

Syntaxcvdwa 12975 The arithmetic progression function.
AP

Syntaxcvdwm 12976 The monochromatic arithmetic progression predicate.
MonoAP

Syntaxcvdwp 12977 The polychromatic arithmetic progression predicate.
PolyAP

Definitiondf-vdwap 12978* Define the arithmetic progression function, which takes as input a length , a start point , and a step and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Definitiondf-vdwmc 12979* Define the "contains a monochromatic AP" predicate. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Definitiondf-vdwpc 12980* Define the "contains a polychromatic colleciton of APs" predicate. See vdwpc 12990 for more information. (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP AP

Theoremvdwapfval 12981* Define the arithmetic progression function, which takes as input a length , a start point , and a step and outputs the set of points in this progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapf 12982 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapval 12983* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapun 12984 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
AP AP

Theoremvdwapid1 12985 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP

Theoremvdwap0 12986 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwap1 12987 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwmc 12988* The predicate " The -coloring contains a monochromatic AP of length ". (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Theoremvdwmc2 12989* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwpc 12990* The predicate " The coloring contains a polychromatic -tuple of AP's of length ". A polychromatic -tuple of AP's is a set of AP's with the same base point but different step lengths, such that each individual AP is monochromatic, but the AP's all have mutually distinct colors. (The common basepoint is not required to have the same color as any of the AP's.) (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP AP

Theoremvdwlem1 12991* Lemma for vdw 13004. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      MonoAP

Theoremvdwlem2 12992* Lemma for vdw 13004. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP MonoAP

Theoremvdwlem3 12993 Lemma for vdw 13004. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwlem4 12994* Lemma for vdw 13004. (Contributed by Mario Carneiro, 12-Sep-2014.)

Theoremvdwlem5 12995* Lemma for vdw 13004. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      AP

Theoremvdwlem6 12996* Lemma for vdw 13004. (Contributed by Mario Carneiro, 13-Sep-2014.)
AP                      AP                                    PolyAP MonoAP

Theoremvdwlem7 12997* Lemma for vdw 13004. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP        PolyAP PolyAP MonoAP

Theoremvdwlem8 12998* Lemma for vdw 13004. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP               PolyAP

Theoremvdwlem9 12999* Lemma for vdw 13004. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP                      PolyAP MonoAP               MonoAP                      PolyAP MonoAP

Theoremvdwlem10 13000* Lemma for vdw 13004. Set up secondary induction on . (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP               PolyAP MonoAP

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