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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremqexpz 14401 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 14402 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.)

6.2.7  Pocklington's theorem

Theoremprmpwdvds 14403 A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthlem 14404 Lemma for pockthg 14405. (Contributed by Mario Carneiro, 2-Mar-2014.)

Theorempockthg 14405* 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 14406 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 14405 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)

6.2.8  Infinite primes theorem

Theoremunbenlem 14407* Lemma for unben 14408. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoremunben 14408* An unbounded set of positive integers is infinite. (Contributed by NM, 5-May-2005.) (Revised by Mario Carneiro, 15-Sep-2013.)

Theoreminfpnlem1 14409* Lemma for infpn 14411. The smallest divisor (greater than 1) of is a prime greater than . (Contributed by NM, 5-May-2005.)

Theoreminfpnlem2 14410* Lemma for infpn 14411. For any positive integer , there exists a prime number greater than . (Contributed by NM, 5-May-2005.)

Theoreminfpn 14411* There exist infinitely many prime numbers: for any positive integer , there exists a prime number greater than . (See infpn2 14412 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)

Theoreminfpn2 14412* There exist infinitely many prime numbers: the set of all primes is unbounded by infpn 14411, so by unben 14408 it is infinite. This is Metamath 100 proof #11. (Contributed by NM, 5-May-2005.)

Theoremprmunb 14413* The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)

Theoremprminf 14414 There are an infinite number of primes. (Contributed by Paul Chapman, 28-Nov-2012.)

6.2.9  Sum of prime reciprocals

Theoremprmreclem1 14415* Lemma for prmrec 14421. Properties of the "square part" function, which extracts the of the decomposition , with maximal and squarefree. (Contributed by Mario Carneiro, 5-Aug-2014.)

Theoremprmreclem2 14416* Lemma for prmrec 14421. 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 14417* Lemma for prmrec 14421. 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 14418* Lemma for prmrec 14421. Show by induction that the indexed (nondisjoint) union is at most the size of the prime reciprocal series. The key counting lemma is hashdvds 14286, to show that the number of numbers in that divide is at most . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmreclem5 14419* Lemma for prmrec 14421. 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 14418 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 14420* Lemma for prmrec 14421. 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 14419 to produce the contradictory bound , which is false for . (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremprmrec 14421* 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. This is Metamath 100 proof #81. (Contributed by Mario Carneiro, 6-Aug-2014.)

6.2.10  Fundamental theorem of arithmetic

Theorem1arithlem1 14422* Lemma for 1arith 14426. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem2 14423* Lemma for 1arith 14426. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem3 14424* Lemma for 1arith 14426. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arithlem4 14425* Lemma for 1arith 14426. (Contributed by Mario Carneiro, 30-May-2014.)

Theorem1arith 14426* 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 14427* 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. This is Metamath 100 proof #80. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.)

6.2.11  Lagrange's four-square theorem

Syntaxcgz 14428 Extend class notation with the set of gaussian integers.

Definitiondf-gz 14429 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 14430 Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcn 14431 A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremzgz 14432 An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremigz 14433 is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgznegcl 14434 The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzcjcl 14435 The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzaddcl 14436 The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzmulcl 14437 The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzreim 14438 Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theoremgzsubcl 14439 The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theoremgzabssqcl 14440 The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem5 14441 Lemma for 4sq 14463. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem6 14442 Lemma for 4sq 14463. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem7 14443 Lemma for 4sq 14463. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem8 14444 Lemma for 4sq 14463. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem9 14445 Lemma for 4sq 14463. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem10 14446 Lemma for 4sq 14463. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem1 14447* Lemma for 4sq 14463. 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 14448* Lemma for 4sq 14463. Change bound variables in . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem3 14449* Lemma for 4sq 14463. Sufficient condition to be in . (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem4a 14450* Lemma for 4sqlem4 14451. (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem4 14451* Lemma for 4sq 14463. 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 14452* Lemma for mul4sq 14453: algebraic manipulations. The extra assumptions involving are for a part of 4sqlem17 14460 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 14453* 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 14452. (For the curious, the explicit formula that is used is .) (Contributed by Mario Carneiro, 14-Jul-2014.)

Theorem4sqlem11 14454* Lemma for 4sq 14463. Use the pigeonhole principle to show that the sets and have a common element, . (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem12 14455* Lemma for 4sq 14463. For any odd prime , there is a such that is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)

Theorem4sqlem13 14456* Lemma for 4sq 14463. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem14 14457* Lemma for 4sq 14463. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem15 14458* Lemma for 4sq 14463. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem16 14459* Lemma for 4sq 14463. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem17 14460* Lemma for 4sq 14463. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem18 14461* Lemma for 4sq 14463. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.)

Theorem4sqlem19 14462* Lemma for 4sq 14463. 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 14461. 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 14453 . (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)

Theorem4sq 14463* Lagrange's four-square theorem, or Bachet's conjecture: every nonnegative integer is expressible as a sum of four squares. This is Metamath 100 proof #19. (Contributed by Mario Carneiro, 16-Jul-2014.)

6.2.12  Van der Waerden's theorem

Syntaxcvdwa 14464 The arithmetic progression function.
AP

Syntaxcvdwm 14465 The monochromatic arithmetic progression predicate.
MonoAP

Syntaxcvdwp 14466 The polychromatic arithmetic progression predicate.
PolyAP

Definitiondf-vdwap 14467* 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 14468* Define the "contains a monochromatic AP" predicate. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Definitiondf-vdwpc 14469* Define the "contains a polychromatic collection of APs" predicate. See vdwpc 14479 for more information. (Contributed by Mario Carneiro, 18-Aug-2014.)
PolyAP AP

Theoremvdwapfval 14470* 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 14471 The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapval 14472* Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwapun 14473 Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.)
AP AP

Theoremvdwapid1 14474 The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP

Theoremvdwap0 14475 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwap1 14476 Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP

Theoremvdwmc 14477* The predicate " The -coloring contains a monochromatic AP of length ". (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP AP

Theoremvdwmc2 14478* Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwpc 14479* 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 14480* Lemma for vdw 14493. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      MonoAP

Theoremvdwlem2 14481* Lemma for vdw 14493. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP MonoAP

Theoremvdwlem3 14482 Lemma for vdw 14493. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwlem4 14483* Lemma for vdw 14493. (Contributed by Mario Carneiro, 12-Sep-2014.)

Theoremvdwlem5 14484* Lemma for vdw 14493. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP                      AP

Theoremvdwlem6 14485* Lemma for vdw 14493. (Contributed by Mario Carneiro, 13-Sep-2014.)
AP                      AP                                    PolyAP MonoAP

Theoremvdwlem7 14486* Lemma for vdw 14493. (Contributed by Mario Carneiro, 12-Sep-2014.)
AP        PolyAP PolyAP MonoAP

Theoremvdwlem8 14487* Lemma for vdw 14493. (Contributed by Mario Carneiro, 18-Aug-2014.)
AP               PolyAP

Theoremvdwlem9 14488* Lemma for vdw 14493. (Contributed by Mario Carneiro, 12-Sep-2014.)
MonoAP                      PolyAP MonoAP               MonoAP                      PolyAP MonoAP

Theoremvdwlem10 14489* Lemma for vdw 14493. Set up secondary induction on . (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP               PolyAP MonoAP

Theoremvdwlem11 14490* Lemma for vdw 14493. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP        MonoAP

Theoremvdwlem12 14491 Lemma for vdw 14493. base case of induction. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdwlem13 14492* Lemma for vdw 14493. Main induction on ; , base cases. (Contributed by Mario Carneiro, 18-Aug-2014.)
MonoAP

Theoremvdw 14493* Van der Waerden's theorem. For any finite coloring and integer , there is an such that every coloring function from to contains a monochromatic arithmetic progression (which written out in full means that there is a color and base, increment values such that all the numbers lie in the preimage of , i.e. they are all in and evaluated at each one yields ). (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem1 14494* Corollary of vdw 14493, and lemma for vdwnn 14497. If is a coloring of the integers, then there are arbitrarily long monochromatic APs in . (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem2 14495* Lemma for vdwnn 14497. The set of all "bad" for the theorem is upwards-closed, because a long AP implies a short AP. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnnlem3 14496* Lemma for vdwnn 14497. (Contributed by Mario Carneiro, 13-Sep-2014.)

Theoremvdwnn 14497* Van der Waerden's theorem, infinitary version. For any finite coloring of the positive integers, there is a color that contains arbitrarily long arithmetic progressions. (Contributed by Mario Carneiro, 13-Sep-2014.)

6.2.13  Ramsey's theorem

Syntaxcram 14498 Extend class notation with the Ramsey number function.
Ramsey

Theoremramtlecl 14499* The set of numbers with the Ramsey number property is upward-closed. (Contributed by Mario Carneiro, 21-Apr-2015.)

Definitiondf-ram 14500* Define the Ramsey number function. The input is a number for the size of the edges of the hypergraph, and a tuple from the finite color set to lower bounds for each color. The Ramsey number Ramsey is the smallest number such that for any set with Ramsey and any coloring of the set of -element subsets of (with color set ), there is a color and a subset such that and all the hyperedges of (that is, subsets of of size ) have color . (Contributed by Mario Carneiro, 20-Apr-2015.)
Ramsey

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