HomeHome Metamath Proof Explorer
Theorem List (p. 171 of 485)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-30800)
  Hilbert Space Explorer  Hilbert Space Explorer
(30801-32323)
  Users' Mathboxes  Users' Mathboxes
(32324-48421)
 

Theorem List for Metamath Proof Explorer - 17001-17100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremramcl 17001 Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.)
((𝑀 ∈ ℕ0𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → (𝑀 Ramsey 𝐹) ∈ ℕ0)
 
Theoremramsey 17002* Ramsey's theorem with the definition of Ramsey (df-ram 16973) eliminated. If 𝑀 is an integer, 𝑅 is a specified finite set of colors, and 𝐹:𝑅⟶ℕ0 is a set of lower bounds for each color, then there is an 𝑛 such that for every set 𝑠 of size greater than 𝑛 and every coloring 𝑓 of the set (𝑠𝐶𝑀) of all 𝑀-element subsets of 𝑠, there is a color 𝑐 and a subset 𝑥𝑠 such that 𝑥 is larger than 𝐹(𝑐) and the 𝑀 -element subsets of 𝑥 are monochromatic with color 𝑐. This is the hypergraph version of Ramsey's theorem; the version for simple graphs is the case 𝑀 = 2. This is Metamath 100 proof #31. (Contributed by Mario Carneiro, 23-Apr-2015.)
𝐶 = (𝑎 ∈ V, 𝑖 ∈ ℕ0 ↦ {𝑏 ∈ 𝒫 𝑎 ∣ (♯‘𝑏) = 𝑖})       ((𝑀 ∈ ℕ0𝑅 ∈ Fin ∧ 𝐹:𝑅⟶ℕ0) → ∃𝑛 ∈ ℕ0𝑠(𝑛 ≤ (♯‘𝑠) → ∀𝑓 ∈ (𝑅m (𝑠𝐶𝑀))∃𝑐𝑅𝑥 ∈ 𝒫 𝑠((𝐹𝑐) ≤ (♯‘𝑥) ∧ (𝑥𝐶𝑀) ⊆ (𝑓 “ {𝑐}))))
 
6.2.15  Primorial function

According to Wikipedia "Primorial", https://en.wikipedia.org/wiki/Primorial (28-Aug-2020): "In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying [all] positive integers [less than or equal to a given number], the function only multiplies [the] prime numbers [less than or equal to the given number]. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors."

 
Syntaxcprmo 17003 Extend class notation to include the primorial of nonnegative integers.
class #p
 
Definitiondf-prmo 17004* Define the primorial function on nonnegative integers as the product of all prime numbers less than or equal to the integer. For example, (#p‘10) = 2 · 3 · 5 · 7 = 210 (see ex-prmo 30341).

In the literature, the primorial function is written as a postscript hash: 6# = 30. In contrast to prmorcht 27155, where the primorial function is defined by using the sequence builder (𝑃 = seq1( · , 𝐹)), the more specialized definition of a product of a series is used here. (Contributed by AV, 28-Aug-2020.)

#p = (𝑛 ∈ ℕ0 ↦ ∏𝑘 ∈ (1...𝑛)if(𝑘 ∈ ℙ, 𝑘, 1))
 
Theoremprmoval 17005* Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) = ∏𝑘 ∈ (1...𝑁)if(𝑘 ∈ ℙ, 𝑘, 1))
 
Theoremprmocl 17006 Closure of the primorial function. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ∈ ℕ)
 
Theoremprmone0 17007 The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≠ 0)
 
Theoremprmo0 17008 The primorial of 0. (Contributed by AV, 28-Aug-2020.)
(#p‘0) = 1
 
Theoremprmo1 17009 The primorial of 1. (Contributed by AV, 28-Aug-2020.)
(#p‘1) = 1
 
Theoremprmop1 17010 The primorial of a successor. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p‘(𝑁 + 1)) = if((𝑁 + 1) ∈ ℙ, ((#p𝑁) · (𝑁 + 1)), (#p𝑁)))
 
Theoremprmonn2 17011 Value of the primorial function expressed recursively. (Contributed by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ → (#p𝑁) = if(𝑁 ∈ ℙ, ((#p‘(𝑁 − 1)) · 𝑁), (#p‘(𝑁 − 1))))
 
Theoremprmo2 17012 The primorial of 2. (Contributed by AV, 28-Aug-2020.)
(#p‘2) = 2
 
Theoremprmo3 17013 The primorial of 3. (Contributed by AV, 28-Aug-2020.)
(#p‘3) = 6
 
Theoremprmdvdsprmo 17014* The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
(𝑁 ∈ ℕ → ∀𝑝 ∈ ℙ (𝑝𝑁𝑝 ∥ (#p𝑁)))
 
Theoremprmdvdsprmop 17015* The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → ∃𝑝 ∈ ℙ (𝑝𝑁𝑝𝐼𝑝 ∥ ((#p𝑁) + 𝐼)))
 
Theoremfvprmselelfz 17016* The value of the prime selection function is in a finite sequence of integers if the argument is in this finite sequence of integers. (Contributed by AV, 19-Aug-2020.)
𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))       ((𝑁 ∈ ℕ ∧ 𝑋 ∈ (1...𝑁)) → (𝐹𝑋) ∈ (1...𝑁))
 
Theoremfvprmselgcd1 17017* The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.)
𝐹 = (𝑚 ∈ ℕ ↦ if(𝑚 ∈ ℙ, 𝑚, 1))       ((𝑋 ∈ (1...𝑁) ∧ 𝑌 ∈ (1...𝑁) ∧ 𝑋𝑌) → ((𝐹𝑋) gcd (𝐹𝑌)) = 1)
 
Theoremprmolefac 17018 The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≤ (!‘𝑁))
 
Theoremprmodvdslcmf 17019 The primorial of a nonnegative integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ∥ (lcm‘(1...𝑁)))
 
Theoremprmolelcmf 17020 The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.)
(𝑁 ∈ ℕ0 → (#p𝑁) ≤ (lcm‘(1...𝑁)))
 
6.2.16  Prime gaps

According to Wikipedia "Prime gap", https://en.wikipedia.org/wiki/Prime_gap (16-Aug-2020): "A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n+1)-th and the n-th prime numbers, i.e. gn = pn+1 - pn . We have g1 = 1, g2 = g3 = 2, and g4 = 4."

It can be proven that there are arbitrary large gaps, usually by showing that "in the sequence n!+2, n!+3, ..., n!+n the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n-1 consecutive composite integers, and it must belong to a gap between primes having length at least n.", see prmgap 17031.

Instead of using the factorial of n (see df-fac 14269), any function can be chosen for which f(n) is not relatively prime to the integers 2, 3, ..., n. For example, the least common multiple of the integers 2, 3, ..., n, see prmgaplcm 17032, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 17034, are such functions, which provide smaller values than the factorial function (see lcmflefac 16622 and prmolefac 17018 resp. prmolelcmf 17020): "For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000." But the least common multiple of the integers 2, 3, ..., 15 is 360360, and 15# is 30030 (p3248 = 30029 and P3249 = 30047, so g3248 = 18).

 
Theoremprmgaplem1 17021 Lemma for prmgap 17031: The factorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((!‘𝑁) + 𝐼))
 
Theoremprmgaplem2 17022 Lemma for prmgap 17031: The factorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 13-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((!‘𝑁) + 𝐼) gcd 𝐼))
 
Theoremprmgaplcmlem1 17023 Lemma for prmgaplcm 17032: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 𝐼 ∥ ((lcm‘(1...𝑁)) + 𝐼))
 
Theoremprmgaplcmlem2 17024 Lemma for prmgaplcm 17032: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((lcm‘(1...𝑁)) + 𝐼) gcd 𝐼))
 
Theoremprmgaplem3 17025* Lemma for prmgap 17031. (Contributed by AV, 9-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ 𝑝 < 𝑁}       (𝑁 ∈ (ℤ‘3) → ∃𝑥𝐴𝑦𝐴 𝑦𝑥)
 
Theoremprmgaplem4 17026* Lemma for prmgap 17031. (Contributed by AV, 10-Aug-2020.)
𝐴 = {𝑝 ∈ ℙ ∣ (𝑁 < 𝑝𝑝𝑃)}       ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑁 < 𝑃) → ∃𝑥𝐴𝑦𝐴 𝑥𝑦)
 
Theoremprmgaplem5 17027* Lemma for prmgap 17031: for each integer greater than 2 there is a smaller prime closest to this integer, i.e. there is a smaller prime and no other prime is between this prime and the integer. (Contributed by AV, 9-Aug-2020.)
(𝑁 ∈ (ℤ‘3) → ∃𝑝 ∈ ℙ (𝑝 < 𝑁 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑁)𝑧 ∉ ℙ))
 
Theoremprmgaplem6 17028* Lemma for prmgap 17031: for each positive integer there is a greater prime closest to this integer, i.e. there is a greater prime and no other prime is between this prime and the integer. (Contributed by AV, 10-Aug-2020.)
(𝑁 ∈ ℕ → ∃𝑝 ∈ ℙ (𝑁 < 𝑝 ∧ ∀𝑧 ∈ ((𝑁 + 1)..^𝑝)𝑧 ∉ ℙ))
 
Theoremprmgaplem7 17029* Lemma for prmgap 17031. (Contributed by AV, 12-Aug-2020.) (Proof shortened by AV, 10-Jul-2022.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑m ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑝 < ((𝐹𝑁) + 2) ∧ ((𝐹𝑁) + 𝑁) < 𝑞 ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
 
Theoremprmgaplem8 17030* Lemma for prmgap 17031. (Contributed by AV, 13-Aug-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐹 ∈ (ℕ ↑m ℕ))    &   (𝜑 → ∀𝑖 ∈ (2...𝑁)1 < (((𝐹𝑁) + 𝑖) gcd 𝑖))       (𝜑 → ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑁 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ))
 
Theoremprmgap 17031* The prime gap theorem: for each positive integer there are (at least) two successive primes with a difference ("gap") at least as big as the given integer. (Contributed by AV, 13-Aug-2020.)
𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)
 
Theoremprmgaplcm 17032* Alternate proof of prmgap 17031: in contrast to prmgap 17031, where the gap starts at n! , the factorial of n, the gap starts at the least common multiple of all positive integers less than or equal to n. (Contributed by AV, 13-Aug-2020.) (Revised by AV, 27-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)
 
Theoremprmgapprmolem 17033 Lemma for prmgapprmo 17034: The primorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.)
((𝑁 ∈ ℕ ∧ 𝐼 ∈ (2...𝑁)) → 1 < (((#p𝑁) + 𝐼) gcd 𝐼))
 
Theoremprmgapprmo 17034* Alternate proof of prmgap 17031: in contrast to prmgap 17031, where the gap starts at n! , the factorial of n, the gap starts at n#, the primorial of n. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑛 ∈ ℕ ∃𝑝 ∈ ℙ ∃𝑞 ∈ ℙ (𝑛 ≤ (𝑞𝑝) ∧ ∀𝑧 ∈ ((𝑝 + 1)..^𝑞)𝑧 ∉ ℙ)
 
6.2.17  Decimal arithmetic (cont.)
 
Theoremdec2dvds 17035 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   (𝐵 · 2) = 𝐶    &   𝐷 = (𝐶 + 1)        ¬ 2 ∥ 𝐴𝐷
 
Theoremdec5dvds 17036 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ    &   𝐵 < 5        ¬ 5 ∥ 𝐴𝐵
 
Theoremdec5dvds2 17037 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ    &   𝐵 < 5    &   (5 + 𝐵) = 𝐶        ¬ 5 ∥ 𝐴𝐶
 
Theoremdec5nprm 17038 Divisibility by five is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ        ¬ 𝐴5 ∈ ℙ
 
Theoremdec2nprm 17039 Divisibility by two is obvious in base 10. (Contributed by Mario Carneiro, 19-Apr-2015.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   (𝐵 · 2) = 𝐶        ¬ 𝐴𝐶 ∈ ℙ
 
Theoremmodxai 17040 Add exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.) (Revised by Mario Carneiro, 5-Feb-2015.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   ((𝐴𝐶) mod 𝑁) = (𝐿 mod 𝑁)    &   (𝐵 + 𝐶) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐿)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)
 
Theoremmod2xi 17041 Double exponents in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   (2 · 𝐵) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)
 
Theoremmodxp1i 17042 Add one to an exponent in a power mod calculation. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐾 mod 𝑁)    &   (𝐵 + 1) = 𝐸    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐴)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)
 
Theoremmod2xnegi 17043 Version of mod2xi 17041 with a negative mod value. (Contributed by Mario Carneiro, 21-Feb-2014.)
𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℤ    &   𝐾 ∈ ℕ    &   𝑀 ∈ ℕ0    &   𝐿 ∈ ℕ0    &   ((𝐴𝐵) mod 𝑁) = (𝐿 mod 𝑁)    &   (2 · 𝐵) = 𝐸    &   (𝐿 + 𝐾) = 𝑁    &   ((𝐷 · 𝑁) + 𝑀) = (𝐾 · 𝐾)       ((𝐴𝐸) mod 𝑁) = (𝑀 mod 𝑁)
 
Theoremmodsubi 17044 Subtract from within a mod calculation. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ    &   𝐴 ∈ ℕ    &   𝐵 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐴 mod 𝑁) = (𝐾 mod 𝑁)    &   (𝑀 + 𝐵) = 𝐾       ((𝐴𝐵) mod 𝑁) = (𝑀 mod 𝑁)
 
Theoremgcdi 17045 Calculate a GCD via Euclid's algorithm. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝐾 ∈ ℕ0    &   𝑅 ∈ ℕ0    &   𝑁 ∈ ℕ0    &   (𝑁 gcd 𝑅) = 𝐺    &   ((𝐾 · 𝑁) + 𝑅) = 𝑀       (𝑀 gcd 𝑁) = 𝐺
 
Theoremgcdmodi 17046 Calculate a GCD via Euclid's algorithm. Theorem 5.6 in [ApostolNT] p. 109. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝐾 ∈ ℕ0    &   𝑅 ∈ ℕ0    &   𝑁 ∈ ℕ    &   (𝐾 mod 𝑁) = (𝑅 mod 𝑁)    &   (𝑁 gcd 𝑅) = 𝐺       (𝐾 gcd 𝑁) = 𝐺
 
Theoremdecexp2 17047 Calculate a power of two. (Contributed by Mario Carneiro, 19-Feb-2014.)
𝑀 ∈ ℕ0    &   (𝑀 + 2) = 𝑁       ((4 · (2↑𝑀)) + 0) = (2↑𝑁)
 
Theoremnumexp0 17048 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (𝐴↑0) = 1
 
Theoremnumexp1 17049 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0       (𝐴↑1) = 𝐴
 
Theoremnumexpp1 17050 Calculate an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝑀 + 1) = 𝑁    &   ((𝐴𝑀) · 𝐴) = 𝐶       (𝐴𝑁) = 𝐶
 
Theoremnumexp2x 17051 Double an integer power. (Contributed by Mario Carneiro, 17-Apr-2015.)
𝐴 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (2 · 𝑀) = 𝑁    &   (𝐴𝑀) = 𝐷    &   (𝐷 · 𝐷) = 𝐶       (𝐴𝑁) = 𝐶
 
Theoremdecsplit0b 17052 Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑0)) + 𝐵) = (𝐴 + 𝐵)
 
Theoremdecsplit0 17053 Split a decimal number into two parts. Base case: 𝑁 = 0. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑0)) + 0) = 𝐴
 
Theoremdecsplit1 17054 Split a decimal number into two parts. Base case: 𝑁 = 1. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0       ((𝐴 · (10↑1)) + 𝐵) = 𝐴𝐵
 
Theoremdecsplit 17055 Split a decimal number into two parts. Inductive step. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 8-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝑀 + 1) = 𝑁    &   ((𝐴 · (10↑𝑀)) + 𝐵) = 𝐶       ((𝐴 · (10↑𝑁)) + 𝐵𝐷) = 𝐶𝐷
 
Theoremkaratsuba 17056 The Karatsuba multiplication algorithm. If 𝑋 and 𝑌 are decomposed into two groups of digits of length 𝑀 (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 𝑋𝑌 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 12775. (Contributed by Mario Carneiro, 16-Jul-2015.) (Revised by AV, 9-Sep-2021.)
𝐴 ∈ ℕ0    &   𝐵 ∈ ℕ0    &   𝐶 ∈ ℕ0    &   𝐷 ∈ ℕ0    &   𝑆 ∈ ℕ0    &   𝑀 ∈ ℕ0    &   (𝐴 · 𝐶) = 𝑅    &   (𝐵 · 𝐷) = 𝑇    &   ((𝐴 + 𝐵) · (𝐶 + 𝐷)) = ((𝑅 + 𝑆) + 𝑇)    &   ((𝐴 · (10↑𝑀)) + 𝐵) = 𝑋    &   ((𝐶 · (10↑𝑀)) + 𝐷) = 𝑌    &   ((𝑅 · (10↑𝑀)) + 𝑆) = 𝑊    &   ((𝑊 · (10↑𝑀)) + 𝑇) = 𝑍       (𝑋 · 𝑌) = 𝑍
 
Theorem2exp4 17057 Two to the fourth power is 16. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑4) = 16
 
Theorem2exp5 17058 Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
(2↑5) = 32
 
Theorem2exp6 17059 Two to the sixth power is 64. (Contributed by Mario Carneiro, 20-Apr-2015.) (Proof shortened by OpenAI, 25-Mar-2020.)
(2↑6) = 64
 
Theorem2exp7 17060 Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
(2↑7) = 128
 
Theorem2exp8 17061 Two to the eighth power is 256. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑8) = 256
 
Theorem2exp11 17062 Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
(2↑11) = 2048
 
Theorem2exp16 17063 Two to the sixteenth power is 65536. (Contributed by Mario Carneiro, 20-Apr-2015.)
(2↑16) = 65536
 
Theorem3exp3 17064 Three to the third power is 27. (Contributed by Mario Carneiro, 20-Apr-2015.)
(3↑3) = 27
 
Theorem2expltfac 17065 The factorial grows faster than two to the power 𝑁. (Contributed by Mario Carneiro, 15-Sep-2016.)
(𝑁 ∈ (ℤ‘4) → (2↑𝑁) < (!‘𝑁))
 
6.2.18  Cyclical shifts of words (cont.)
 
Theoremcshwsidrepsw 17066 If cyclically shifting a word of length being a prime number by a number of positions which is not divisible by the prime number results in the word itself, the word is a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((𝐿 ∈ ℤ ∧ (𝐿 mod (♯‘𝑊)) ≠ 0 ∧ (𝑊 cyclShift 𝐿) = 𝑊) → 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊))))
 
Theoremcshwsidrepswmod0 17067 If cyclically shifting a word of length being a prime number results in the word itself, the shift must be either by 0 (modulo the length of the word) or the word must be a "repeated symbol word". (Contributed by AV, 18-May-2018.) (Revised by AV, 10-Nov-2018.)
((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ ∧ 𝐿 ∈ ℤ) → ((𝑊 cyclShift 𝐿) = 𝑊 → ((𝐿 mod (♯‘𝑊)) = 0 ∨ 𝑊 = ((𝑊‘0) repeatS (♯‘𝑊)))))
 
Theoremcshwshashlem1 17068* If cyclically shifting a word of length being a prime number not consisting of identical symbols by at least one position (and not by as many positions as the length of the word), the result will not be the word itself. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Revised by AV, 10-Nov-2018.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0) ∧ 𝐿 ∈ (1..^(♯‘𝑊))) → (𝑊 cyclShift 𝐿) ≠ 𝑊)
 
Theoremcshwshashlem2 17069* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 < 𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
 
Theoremcshwshashlem3 17070* If cyclically shifting a word of length being a prime number and not of identical symbols by different numbers of positions, the resulting words are different. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0)) → ((𝐿 ∈ (0..^(♯‘𝑊)) ∧ 𝐾 ∈ (0..^(♯‘𝑊)) ∧ 𝐾𝐿) → (𝑊 cyclShift 𝐿) ≠ (𝑊 cyclShift 𝐾)))
 
Theoremcshwsdisj 17071* The singletons resulting by cyclically shifting a given word of length being a prime number and not consisting of identical symbols is a disjoint collection. (Contributed by Alexander van der Vekens, 19-May-2018.) (Revised by Alexander van der Vekens, 8-Jun-2018.)
(𝜑 → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ))       ((𝜑 ∧ ∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0)) → Disj 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
 
Theoremcshwsiun 17072* The set of (different!) words resulting by cyclically shifting a given word is an indexed union. (Contributed by AV, 19-May-2018.) (Revised by AV, 8-Jun-2018.) (Proof shortened by AV, 8-Nov-2018.)
𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       (𝑊 ∈ Word 𝑉𝑀 = 𝑛 ∈ (0..^(♯‘𝑊)){(𝑊 cyclShift 𝑛)})
 
Theoremcshwsex 17073* The class of (different!) words resulting by cyclically shifting a given word is a set. (Contributed by AV, 8-Jun-2018.) (Revised by AV, 8-Nov-2018.)
𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       (𝑊 ∈ Word 𝑉𝑀 ∈ V)
 
Theoremcshws0 17074* The size of the set of (different!) words resulting by cyclically shifting an empty word is 0. (Contributed by AV, 8-Nov-2018.)
𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       (𝑊 = ∅ → (♯‘𝑀) = 0)
 
Theoremcshwrepswhash1 17075* The size of the set of (different!) words resulting by cyclically shifting a nonempty "repeated symbol word" is 1. (Contributed by AV, 18-May-2018.) (Revised by AV, 8-Nov-2018.)
𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       ((𝐴𝑉𝑁 ∈ ℕ ∧ 𝑊 = (𝐴 repeatS 𝑁)) → (♯‘𝑀) = 1)
 
Theoremcshwshashnsame 17076* If a word (not consisting of identical symbols) has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → (∃𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) ≠ (𝑊‘0) → (♯‘𝑀) = (♯‘𝑊)))
 
Theoremcshwshash 17077* If a word has a length being a prime number, the size of the set of (different!) words resulting by cyclically shifting the original word equals the length of the original word or 1. (Contributed by AV, 19-May-2018.) (Revised by AV, 10-Nov-2018.)
𝑀 = {𝑤 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0..^(♯‘𝑊))(𝑊 cyclShift 𝑛) = 𝑤}       ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) ∈ ℙ) → ((♯‘𝑀) = (♯‘𝑊) ∨ (♯‘𝑀) = 1))
 
6.2.19  Specific prime numbers
 
Theoremprmlem0 17078* Lemma for prmlem1 17080 and prmlem2 17092. (Contributed by Mario Carneiro, 18-Feb-2014.)
((¬ 2 ∥ 𝑀𝑥 ∈ (ℤ𝑀)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))    &   (𝐾 ∈ ℙ → ¬ 𝐾𝑁)    &   (𝐾 + 2) = 𝑀       ((¬ 2 ∥ 𝐾𝑥 ∈ (ℤ𝐾)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))
 
Theoremprmlem1a 17079* A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ    &   1 < 𝑁    &    ¬ 2 ∥ 𝑁    &    ¬ 3 ∥ 𝑁    &   ((¬ 2 ∥ 5 ∧ 𝑥 ∈ (ℤ‘5)) → ((𝑥 ∈ (ℙ ∖ {2}) ∧ (𝑥↑2) ≤ 𝑁) → ¬ 𝑥𝑁))       𝑁 ∈ ℙ
 
Theoremprmlem1 17080 A quick proof skeleton to show that the numbers less than 25 are prime, by trial division. (Contributed by Mario Carneiro, 18-Feb-2014.)
𝑁 ∈ ℕ    &   1 < 𝑁    &    ¬ 2 ∥ 𝑁    &    ¬ 3 ∥ 𝑁    &   𝑁 < 25       𝑁 ∈ ℙ
 
Theorem5prm 17081 5 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
5 ∈ ℙ
 
Theorem6nprm 17082 6 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 6 ∈ ℙ
 
Theorem7prm 17083 7 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
7 ∈ ℙ
 
Theorem8nprm 17084 8 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 8 ∈ ℙ
 
Theorem9nprm 17085 9 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.)
¬ 9 ∈ ℙ
 
Theorem10nprm 17086 10 is not a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
¬ 10 ∈ ℙ
 
Theorem11prm 17087 11 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
11 ∈ ℙ
 
Theorem13prm 17088 13 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
13 ∈ ℙ
 
Theorem17prm 17089 17 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
17 ∈ ℙ
 
Theorem19prm 17090 19 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
19 ∈ ℙ
 
Theorem23prm 17091 23 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by Mario Carneiro, 20-Apr-2015.)
23 ∈ ℙ
 
Theoremprmlem2 17092 Our last proving session got as far as 25 because we started with the two "bootstrap" primes 2 and 3, and the next prime is 5, so knowing that 2 and 3 are prime and 4 is not allows to cover the numbers less than 5↑2 = 25. Additionally, nonprimes are "easy", so we can extend this range of known prime/nonprimes all the way until 29, which is the first prime larger than 25. Thus, in this lemma we extend another blanket out to 29↑2 = 841, from which we can prove even more primes. If we wanted, we could keep doing this, but the goal is Bertrand's postulate, and for that we only need a few large primes - we don't need to find them all, as we have been doing thus far. So after this blanket runs out, we'll have to switch to another method (see 1259prm 17108).

As a side note, you can see the pattern of the primes in the indentation pattern of this lemma! (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)

𝑁 ∈ ℕ    &   𝑁 < 841    &   1 < 𝑁    &    ¬ 2 ∥ 𝑁    &    ¬ 3 ∥ 𝑁    &    ¬ 5 ∥ 𝑁    &    ¬ 7 ∥ 𝑁    &    ¬ 11 ∥ 𝑁    &    ¬ 13 ∥ 𝑁    &    ¬ 17 ∥ 𝑁    &    ¬ 19 ∥ 𝑁    &    ¬ 23 ∥ 𝑁       𝑁 ∈ ℙ
 
Theorem37prm 17093 37 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
37 ∈ ℙ
 
Theorem43prm 17094 43 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
43 ∈ ℙ
 
Theorem83prm 17095 83 is a prime number. (Contributed by Mario Carneiro, 18-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
83 ∈ ℙ
 
Theorem139prm 17096 139 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
139 ∈ ℙ
 
Theorem163prm 17097 163 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
163 ∈ ℙ
 
Theorem317prm 17098 317 is a prime number. (Contributed by Mario Carneiro, 19-Feb-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
317 ∈ ℙ
 
Theorem631prm 17099 631 is a prime number. (Contributed by Mario Carneiro, 1-Mar-2014.) (Proof shortened by Mario Carneiro, 20-Apr-2015.)
631 ∈ ℙ
 
Theoremprmo4 17100 The primorial of 4. (Contributed by AV, 28-Aug-2020.)
(#p‘4) = 6
    < 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 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-48000 481 48001-48100 482 48101-48200 483 48201-48300 484 48301-48400 485 48401-48421
  Copyright terms: Public domain < Previous  Next >