P. 130 of Theorem List - Intuitionistic Logic Explorer

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**Theorem List for Intuitionistic Logic Explorer -** 12901-13000 *Has distinct variable group(s)
Type	Label	Description
Statement

Theorem	difsqpwdvds 12901	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 12902	Lemma for pcadd 12903. 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 12903	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 12904	The inequality of pcadd 12903 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 12905	Closure for the prime power map. (Contributed by Mario Carneiro, 12-Mar-2014.)
$/\$

Theorem	pcmpt 12906*	Construct a function with given prime count characteristics. (Contributed by Mario Carneiro, 12-Mar-2014.)


Theorem	pcmpt2 12907*	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 12908	The partial products of the prime power map form a divisibility chain. (Contributed by Mario Carneiro, 12-Mar-2014.)


Theorem	pcprod 12909*	The product of the primes taken to their respective powers reconstructs the original number. (Contributed by Mario Carneiro, 12-Mar-2014.)


Theorem	sumhashdc 12910*	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 12911	The difference between the floors of adjacent fractions is either 1 or 0. (Contributed by Mario Carneiro, 8-Mar-2014.)
$/\$

Theorem	pcfaclem 12912	Lemma for pcfac 12913. (Contributed by Mario Carneiro, 20-May-2014.)
$/\$ $/\$

Theorem	pcfac 12913*	Calculate the prime count of a factorial. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
$/\$ $/\$

Theorem	pcbc 12914*	Calculate the prime count of a binomial coefficient. (Contributed by Mario Carneiro, 11-Mar-2014.) (Revised by Mario Carneiro, 21-May-2014.)
$/\$ $/\$

Theorem	qexpz 12915	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 12916	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 12917*	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 12918	A relation involving divisibility by a prime power. (Contributed by Mario Carneiro, 2-Mar-2014.)
$/\$ $/\$ $/\$ $/\$ $/\$

Theorem	pockthlem 12919	Lemma for pockthg 12920. (Contributed by Mario Carneiro, 2-Mar-2014.)


Theorem	pockthg 12920*	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 12921	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 12920 for a more general closed-form version. (Contributed by Mario Carneiro, 2-Mar-2014.)


5.2.10 Infinite primes theorem

Theorem	infpnlem1 12922*	Lemma for infpn 12924. The smallest divisor (greater than 1) of is a prime greater than . (Contributed by NM, 5-May-2005.)
$/\$ $/\$ $/\$ $/\$ $/\$ $\/$

Theorem	infpnlem2 12923*	Lemma for infpn 12924. For any positive integer , there exists a prime number greater than . (Contributed by NM, 5-May-2005.)
$/\$ $\/$

Theorem	infpn 12924*	There exist infinitely many prime numbers: for any positive integer , there exists a prime number greater than . (See infpn2 13067 for the equinumerosity version.) (Contributed by NM, 1-Jun-2006.)
$/\$ $\/$

Theorem	prmunb 12925*	The primes are unbounded. (Contributed by Paul Chapman, 28-Nov-2012.)


5.2.11 Fundamental theorem of arithmetic

Theorem	1arithlem1 12926*	Lemma for 1arith 12930. (Contributed by Mario Carneiro, 30-May-2014.)


Theorem	1arithlem2 12927*	Lemma for 1arith 12930. (Contributed by Mario Carneiro, 30-May-2014.)
$/\$

Theorem	1arithlem3 12928*	Lemma for 1arith 12930. (Contributed by Mario Carneiro, 30-May-2014.)


Theorem	1arithlem4 12929*	Lemma for 1arith 12930. (Contributed by Mario Carneiro, 30-May-2014.)
$/\$ $/\$

Theorem	1arith 12930*	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 12931*	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 12932	Extend class notation with the set of gaussian integers.


Definition	df-gz 12933	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 12934	Elementhood in the gaussian integers. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$

Theorem	gzcn 12935	A gaussian integer is a complex number. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	zgz 12936	An integer is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	igz 12937	is a gaussian integer. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	gznegcl 12938	The gaussian integers are closed under negation. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	gzcjcl 12939	The gaussian integers are closed under conjugation. (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	gzaddcl 12940	The gaussian integers are closed under addition. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	gzmulcl 12941	The gaussian integers are closed under multiplication. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	gzreim 12942	Construct a gaussian integer from real and imaginary parts. (Contributed by Mario Carneiro, 16-Jul-2014.)
$/\$

Theorem	gzsubcl 12943	The gaussian integers are closed under subtraction. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	gzabssqcl 12944	The squared norm of a gaussian integer is an integer. (Contributed by Mario Carneiro, 16-Jul-2014.)


Theorem	4sqlem5 12945	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 15-Jul-2014.)
$/\$

Theorem	4sqlem6 12946	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 15-Jul-2014.)
$/\$

Theorem	4sqlem7 12947	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem8 12948	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem9 12949	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 15-Jul-2014.)
$/\$ $/\$

Theorem	4sqlem10 12950	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 16-Jul-2014.)
$/\$ $/\$

Theorem	4sqlem1 12951*	Lemma for 4sq 12973. 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 12952*	Lemma for 4sq 12973. Change bound variables in . (Contributed by Mario Carneiro, 14-Jul-2014.)


Theorem	4sqlem3 12953*	Lemma for 4sq 12973. Sufficient condition to be in . (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$ $/\$ $/\$

Theorem	4sqlem4a 12954*	Lemma for 4sqlem4 12955. (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	4sqlem4 12955*	Lemma for 4sq 12973. 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 12956*	Lemma for mul4sq 12957: 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 12957*	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 12956. (For the curious, the explicit formula that is used is .) (Contributed by Mario Carneiro, 14-Jul-2014.)
$/\$

Theorem	4sqlemafi 12958*	Lemma for 4sq 12973. is finite. (Contributed by Jim Kingdon, 24-May-2025.)


Theorem	4sqlemffi 12959*	Lemma for 4sq 12973. is finite. (Contributed by Jim Kingdon, 24-May-2025.)


Theorem	4sqleminfi 12960*	Lemma for 4sq 12973. is finite. (Contributed by Jim Kingdon, 24-May-2025.)


Theorem	4sqexercise1 12961*	Exercise which may help in understanding the proof of 4sqlemsdc 12963. (Contributed by Jim Kingdon, 25-May-2025.)
DECID

Theorem	4sqexercise2 12962*	Exercise which may help in understanding the proof of 4sqlemsdc 12963. (Contributed by Jim Kingdon, 30-May-2025.)
DECID

Theorem	4sqlemsdc 12963*	Lemma for 4sq 12973. 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 12961 and 4sqexercise2 12962 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 12964*	Lemma for 4sq 12973. 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 12960 and fin0 7067. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem12 12965*	Lemma for 4sq 12973. For any odd prime , there is a such that is a sum of two squares. (Contributed by Mario Carneiro, 15-Jul-2014.)


Theorem	4sqlem13m 12966*	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf $/\$

Theorem	4sqlem14 12967*	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf

Theorem	4sqlem15 12968*	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf $/\$ $/\$ $/\$ $/\$

Theorem	4sqlem16 12969*	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf $/\$ $\/$

Theorem	4sqlem17 12970*	Lemma for 4sq 12973. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf

Theorem	4sqlem18 12971*	Lemma for 4sq 12973. Inductive step, odd prime case. (Contributed by Mario Carneiro, 16-Jul-2014.) (Revised by AV, 14-Sep-2020.)
inf

Theorem	4sqlem19 12972*	Lemma for 4sq 12973. 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 12971. 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 12957 . (Contributed by Mario Carneiro, 14-Jul-2014.) (Revised by Mario Carneiro, 20-Jun-2015.)


Theorem	4sq 12973*	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 12974	Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec5dvds 12975	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec5dvds2 12976	Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec5nprm 12977	A decimal number greater than 10 and ending with five is not a prime number. (Contributed by Mario Carneiro, 19-Apr-2015.)
;

Theorem	dec2nprm 12978	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 12979	Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)


Theorem	mod2xi 12980	Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)


Theorem	modxp1i 12981	Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)


Theorem	modsubi 12982	Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)


Theorem	gcdi 12983	Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)


Theorem	gcdmodi 12984	Calculate a GCD via Euclid's algorithm. Theorem 5.6 in [ApostolNT] p. 109. (Contributed by Mario Carneiro, 19-Feb-2014.)


Theorem	numexp0 12985	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	numexp1 12986	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	numexpp1 12987	Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	numexp2x 12988	Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)


Theorem	decsplit0b 12989	Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decsplit0 12990	Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
;

Theorem	decsplit1 12991	Split a decimal number into two parts. Base case: . (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
; ;

Theorem	decsplit 12992	Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
; ; ; ;

Theorem	karatsuba 12993	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 9665. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)
; ; ; ;

Theorem	2exp4 12994	Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
;

Theorem	2exp5 12995	Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
;

Theorem	2exp6 12996	Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
;

Theorem	2exp7 12997	Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
;;

Theorem	2exp8 12998	Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
;;

Theorem	2exp11 12999	Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
; ;;;

Theorem	2exp16 13000	Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
; ;;;;

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