P. 127 of Theorem List - Intuitionistic Logic Explorer

Home	Intuitionistic Logic Explorer Theorem List (p. 127 of 160)	< Previous Next >
		Browser slow? Try the Unicode version.
Mirrors > Metamath Home Page > ILE Home Page > Theorem List Contents > Recent Proofs This page: Page List

**Theorem List for Intuitionistic Logic Explorer -** 12601-12700 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	dvdsprmpweqnn 12601*	If an integer greater than 1 divides a prime power, it is a (proper) prime power. (Contributed by AV, 13-Aug-2021.)
$/\$ $/\$

Theorem	dvdsprmpweqle 12602*	If a positive integer divides a prime power, it is a prime power with a smaller exponent. (Contributed by AV, 25-Jul-2021.)
$/\$ $/\$ $/\$

Theorem	difsqpwdvds 12603	If the difference of two squares is a power of a prime, the prime divides twice the second squared number. (Contributed by AV, 13-Aug-2021.)
$/\$ $/\$ $/\$ $/\$

Theorem	pcaddlem 12604	Lemma for pcadd 12605. 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.)
$/\$ $/\$ $/\$ $/\$

Theorem	pcadd 12605	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.)


Theorem	pcadd2 12606	The inequality of pcadd 12605 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.)


Theorem	pcmptcl 12607	Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)
$/\$

Theorem	pcmpt 12608*	Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)


Theorem	pcmpt2 12609*	Dividing two prime count maps yields a number with all dividing primes confined to an interval. (Contributed by Mario Carneiro, 14-Mar-2014.)
$/\$

Theorem	pcmptdvds 12610	The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)


Theorem	pcprod 12611*	The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)


Theorem	sumhashdc 12612*	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.)
$/\$ $/\$ DECID ♯

Theorem	fldivp1 12613	The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)
$/\$

Theorem	pcfaclem 12614	Lemma for pcfac 12615. (Contributed by Mario Carneiro, 20-May-2014.)
$/\$ $/\$

Theorem	pcfac 12615*	Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
$/\$ $/\$

Theorem	pcbc 12616*	Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
$/\$ $/\$

Theorem	qexpz 12617	If a power of a rational number is an integer, then the number is an integer. (Contributed by Mario Carneiro, 10-Aug-2015.)
$/\$ $/\$

Theorem	expnprm 12618	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 not rational. Suggested by Norm Megill. (Contributed by Mario Carneiro, 10-Aug-2015.)
$/\$

Theorem	oddprmdvds 12619*	Every positive integer which is not a power of two is divisible by an odd prime number. (Contributed by AV, 6-Aug-2021.)
$/\$ $\$

5.2.9 Pocklington's theorem

Theorem	prmpwdvds 12620	A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pockthlem 12621	Lemma for pockthg 12622. (Contributed by Mario Carneiro, 2-Mar-2014.)


Theorem	pockthg 12622*	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.)
$/\$

Theorem	pockthi 12623	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 12622 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)


5.2.10 Infinite primes theorem

Theorem	infpnlem1 12624*	Lemma for infpn 12626. The smallest divisor (greater than 1) of is a prime greater than . (Contributed by NM, 5-May-2005.)
$/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Theorem	infpnlem2 12625*	Lemma for infpn 12626. For any positive integer , there exists a prime number greater than . (Contributed by NM, 5-May-2005.)
$/\$ $\/$

Theorem	infpn 12626*	There exist infinitely many prime numbers: for any positive integer , there exists a prime number greater than . (See infpn2 12769 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)
$/\$ $\/$

Theorem	prmunb 12627*	The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)


5.2.11 Fundamental theorem of arithmetic

Theorem	1arithlem1 12628*	Lemma for 1arith 12632. (Contributed by Mario Carneiro, 30-May-2014.)


Theorem	1arithlem2 12629*	Lemma for 1arith 12632. (Contributed by Mario Carneiro, 30-May-2014.)
$/\$

Theorem	1arithlem3 12630*	Lemma for 1arith 12632. (Contributed by Mario Carneiro, 30-May-2014.)


Theorem	1arithlem4 12631*	Lemma for 1arith 12632. (Contributed by Mario Carneiro, 30-May-2014.)
$/\$ $/\$

Theorem	1arith 12632*	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.)


Theorem	1arith2 12633*	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. Theorem 1.10 in [ApostolNT] p. 17. This is Metamath 100 proof #80. (Contributed by Paul Chapman, 17-Nov-2012.) (Revised by Mario Carneiro, 30-May-2014.)


5.2.12 Lagrange's four-square theorem

Syntax	cgz 12634	Extend class notation with the set of gaussian integers.


Definition	df-gz 12635	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.)
$/\$

Theorem	elgz 12636	Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$

Theorem	gzcn 12637	A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	zgz 12638	An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	igz 12639	is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	gznegcl 12640	The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	gzcjcl 12641	The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	gzaddcl 12642	The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	gzmulcl 12643	The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	gzreim 12644	Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)
$/\$

Theorem	gzsubcl 12645	The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	gzabssqcl 12646	The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)


Theorem	4sqlem5 12647	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 15-Jul-2014.)
$/\$

Theorem	4sqlem6 12648	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 15-Jul-2014.)
$/\$

Theorem	4sqlem7 12649	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem8 12650	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem9 12651	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 15-Jul-2014.)
$/\$ $/\$

Theorem	4sqlem10 12652	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 16-Jul-2014.)
$/\$ $/\$

Theorem	4sqlem1 12653*	Lemma for 4sq 12675. 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.)


Theorem	4sqlem2 12654*	Lemma for 4sq 12675. Change bound variables in . (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	4sqlem3 12655*	Lemma for 4sq 12675. Sufficient condition to be in . (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	4sqlem4a 12656*	Lemma for 4sqlem4 12657. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	4sqlem4 12657*	Lemma for 4sq 12675. 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.)


Theorem	mul4sqlem 12658*	Lemma for mul4sq 12659: algebraic manipulations. The extra assumptions involving would let us 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.)


Theorem	mul4sq 12659*	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 12658. (For the curious, the explicit formula that is used is .) (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	4sqlemafi 12660*	Lemma for 4sq 12675. is finite. (Contributed by Jim Kingdon, 24-May-2025.)


Theorem	4sqlemffi 12661*	Lemma for 4sq 12675. is finite. (Contributed by Jim Kingdon, 24-May-2025.)


Theorem	4sqleminfi 12662*	Lemma for 4sq 12675. is finite. (Contributed by Jim Kingdon, 24-May-2025.)


Theorem	4sqexercise1 12663*	Exercise which may help in understanding the proof of 4sqlemsdc 12665. (Contributed by Jim Kingdon, 25-May-2025.)
DECID

Theorem	4sqexercise2 12664*	Exercise which may help in understanding the proof of 4sqlemsdc 12665. (Contributed by Jim Kingdon, 30-May-2025.)
DECID

Theorem	4sqlemsdc 12665*	Lemma for 4sq 12675. The property of being the sum of four squares is decidable. The proof involves showing that (for a particular ) there are only a finite number of possible ways that it could be the sum of four squares, so checking each of those possibilities in turn decides whether the number is the sum of four squares. If this proof is hard to follow, especially because of its length, the simplified versions at 4sqexercise1 12663 and 4sqexercise2 12664 may help clarify, as they are using very much the same techniques on simplified versions of this lemma. (Contributed by Jim Kingdon, 25-May-2025.)
DECID

Theorem	4sqlem11 12666*	Lemma for 4sq 12675. Use the pigeonhole principle to show that the sets and have a common element, . Note that although the conclusion is stated in terms of being nonempty, it is also inhabited by 4sqleminfi 12662 and fin0 6981. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem12 12667*	Lemma for 4sq 12675. For any odd prime , there is a such that is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem13m 12668*	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf $/\$

Theorem	4sqlem14 12669*	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf

Theorem	4sqlem15 12670*	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf $/\$ $/\$ $/\$ $/\$

Theorem	4sqlem16 12671*	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf $/\$ $\/$

Theorem	4sqlem17 12672*	Lemma for 4sq 12675. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf

Theorem	4sqlem18 12673*	Lemma for 4sq 12675. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf

Theorem	4sqlem19 12674*	Lemma for 4sq 12675. 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 12673. 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 12659 . (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)


Theorem	4sq 12675*	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.)


5.2.13 Decimal arithmetic (cont.)

Theorem	dec2dvds 12676	Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec5dvds 12677	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec5dvds2 12678	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec5nprm 12679	A decimal number greater than 10 and ending with five is not a prime number. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec2nprm 12680	A decimal number greater than 10 and ending with an even digit is not a prime number. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	modxai 12681	Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)


Theorem	mod2xi 12682	Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)


Theorem	modxp1i 12683	Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)


Theorem	modsubi 12684	Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	gcdi 12685	Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)


Theorem	gcdmodi 12686	Calculate a GCD via Euclid's algorithm. Theorem 5.6 in [ApostolNT] p. 109. (Contributed by Mario Carneiro, 19-Feb-2014.)


Theorem	numexp0 12687	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	numexp1 12688	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	numexpp1 12689	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	numexp2x 12690	Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	decsplit0b 12691	Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decsplit0 12692	Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decsplit1 12693	Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	decsplit 12694	Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
; ; ; ;

Theorem	karatsuba 12695	The Karatsuba multiplication algorithm. If and are decomposed into two groups of digits of length (only the lower group is known to be this size but the algorithm is most efficient when the partition is chosen near the middle of the digit string), then can be written in three groups of digits, where each group needs only one multiplication. Thus, we can halve both inputs with only three multiplications on the smaller operands, yielding an asymptotic improvement of n^(log₂ 3) instead of n^2 for the "naive" algorithm decmul1c 9567. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)
; ; ; ;

Theorem	2exp4 12696	Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem	2exp5 12697	Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
;

Theorem	2exp6 12698	Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
;

Theorem	2exp7 12699	Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
;;

Theorem	2exp8 12700	Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
;;

< Previous Next >

Page List Jump to page: Contents 1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-15956

< Previous Next >