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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlgsdirprm 26601 The Legendre symbol is completely multiplicative at the primes. See theorem 9.3 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 4-Feb-2015.) (Proof shortened by AV, 18-Mar-2022.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑃 ∈ β„™) β†’ ((𝐴 Β· 𝐡) /L 𝑃) = ((𝐴 /L 𝑃) Β· (𝐡 /L 𝑃)))
 
Theoremlgsdir 26602 The Legendre symbol is completely multiplicative in its left argument. Generalization of theorem 9.9(a) in [ApostolNT] p. 188 (which assumes that 𝐴 and 𝐡 are odd positive integers). Together with lgsqr 26621 this implies that the product of two quadratic residues or nonresidues is a residue, and the product of a residue and a nonresidue is a nonresidue. (Contributed by Mario Carneiro, 4-Feb-2015.)
(((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝐴 β‰  0 ∧ 𝐡 β‰  0)) β†’ ((𝐴 Β· 𝐡) /L 𝑁) = ((𝐴 /L 𝑁) Β· (𝐡 /L 𝑁)))
 
Theoremlgsdilem2 26603* Lemma for lgsdi 26604. (Contributed by Mario Carneiro, 4-Feb-2015.)
(πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝑀 ∈ β„€)    &   (πœ‘ β†’ 𝑁 ∈ β„€)    &   (πœ‘ β†’ 𝑀 β‰  0)    &   (πœ‘ β†’ 𝑁 β‰  0)    &   πΉ = (𝑛 ∈ β„• ↦ if(𝑛 ∈ β„™, ((𝐴 /L 𝑛)↑(𝑛 pCnt 𝑀)), 1))    β‡’   (πœ‘ β†’ (seq1( Β· , 𝐹)β€˜(absβ€˜π‘€)) = (seq1( Β· , 𝐹)β€˜(absβ€˜(𝑀 Β· 𝑁))))
 
Theoremlgsdi 26604 The Legendre symbol is completely multiplicative in its right argument. Generalization of theorem 9.9(b) in [ApostolNT] p. 188 (which assumes that 𝑀 and 𝑁 are odd positive integers). (Contributed by Mario Carneiro, 5-Feb-2015.)
(((𝐴 ∈ β„€ ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) ∧ (𝑀 β‰  0 ∧ 𝑁 β‰  0)) β†’ (𝐴 /L (𝑀 Β· 𝑁)) = ((𝐴 /L 𝑀) Β· (𝐴 /L 𝑁)))
 
Theoremlgsne0 26605 The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((𝐴 /L 𝑁) β‰  0 ↔ (𝐴 gcd 𝑁) = 1))
 
Theoremlgsabs1 26606 The Legendre symbol is nonzero (and hence equal to 1 or -1) precisely when the arguments are coprime. (Contributed by Mario Carneiro, 5-Feb-2015.)
((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ ((absβ€˜(𝐴 /L 𝑁)) = 1 ↔ (𝐴 gcd 𝑁) = 1))
 
Theoremlgssq 26607 The Legendre symbol at a square is equal to 1. Together with lgsmod 26593 this implies that the Legendre symbol takes value 1 at every quadratic residue. (Contributed by Mario Carneiro, 5-Feb-2015.) (Revised by AV, 20-Jul-2021.)
(((𝐴 ∈ β„€ ∧ 𝐴 β‰  0) ∧ 𝑁 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1) β†’ ((𝐴↑2) /L 𝑁) = 1)
 
Theoremlgssq2 26608 The Legendre symbol at a square is equal to 1. (Contributed by Mario Carneiro, 5-Feb-2015.)
((𝐴 ∈ β„€ ∧ 𝑁 ∈ β„• ∧ (𝐴 gcd 𝑁) = 1) β†’ (𝐴 /L (𝑁↑2)) = 1)
 
Theoremlgsprme0 26609 The Legendre symbol at any prime (even at 2) is 0 iff the prime does not divide the first argument. See definition in [ApostolNT] p. 179. (Contributed by AV, 20-Jul-2021.)
((𝐴 ∈ β„€ ∧ 𝑃 ∈ β„™) β†’ ((𝐴 /L 𝑃) = 0 ↔ (𝐴 mod 𝑃) = 0))
 
Theorem1lgs 26610 The Legendre symbol at 1. See example 1 in [ApostolNT] p. 180. (Contributed by Mario Carneiro, 28-Apr-2016.)
(𝑁 ∈ β„€ β†’ (1 /L 𝑁) = 1)
 
Theoremlgs1 26611 The Legendre symbol at 1. See definition in [ApostolNT] p. 188. (Contributed by Mario Carneiro, 28-Apr-2016.)
(𝐴 ∈ β„€ β†’ (𝐴 /L 1) = 1)
 
Theoremlgsmodeq 26612 The Legendre (Jacobi) symbol is preserved under reduction mod 𝑛 when 𝑛 is odd. Theorem 9.9(c) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ (𝑁 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑁)) β†’ ((𝐴 mod 𝑁) = (𝐡 mod 𝑁) β†’ (𝐴 /L 𝑁) = (𝐡 /L 𝑁)))
 
Theoremlgsmulsqcoprm 26613 The Legendre (Jacobi) symbol is preserved under multiplication with a square of an integer coprime to the second argument. Theorem 9.9(d) in [ApostolNT] p. 188. (Contributed by AV, 20-Jul-2021.)
(((𝐴 ∈ β„€ ∧ 𝐴 β‰  0) ∧ (𝐡 ∈ β„€ ∧ 𝐡 β‰  0) ∧ (𝑁 ∈ β„€ ∧ (𝐴 gcd 𝑁) = 1)) β†’ (((𝐴↑2) Β· 𝐡) /L 𝑁) = (𝐡 /L 𝑁))
 
Theoremlgsdirnn0 26614 Variation on lgsdir 26602 valid for all 𝐴, 𝐡 but only for positive 𝑁. (The exact location of the failure of this law is for 𝐴 = 0, 𝐡 < 0, 𝑁 = -1 in which case (0 /L -1) = 1 but (𝐡 /L -1) = -1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑁 ∈ β„•0) β†’ ((𝐴 Β· 𝐡) /L 𝑁) = ((𝐴 /L 𝑁) Β· (𝐡 /L 𝑁)))
 
Theoremlgsdinn0 26615 Variation on lgsdi 26604 valid for all 𝑀, 𝑁 but only for positive 𝐴. (The exact location of the failure of this law is for 𝐴 = -1, 𝑀 = 0, and some 𝑁 in which case (-1 /L 0) = 1 but (-1 /L 𝑁) = -1 when -1 is not a quadratic residue mod 𝑁.) (Contributed by Mario Carneiro, 28-Apr-2016.)
((𝐴 ∈ β„•0 ∧ 𝑀 ∈ β„€ ∧ 𝑁 ∈ β„€) β†’ (𝐴 /L (𝑀 Β· 𝑁)) = ((𝐴 /L 𝑀) Β· (𝐴 /L 𝑁)))
 
Theoremlgsqrlem1 26616 Lemma for lgsqr 26621. (Contributed by Mario Carneiro, 15-Jun-2015.)
π‘Œ = (β„€/nβ„€β€˜π‘ƒ)    &   π‘† = (Poly1β€˜π‘Œ)    &   π΅ = (Baseβ€˜π‘†)    &   π· = ( deg1 β€˜π‘Œ)    &   π‘‚ = (eval1β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   π‘‹ = (var1β€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘†)    &    1 = (1rβ€˜π‘†)    &   π‘‡ = ((((𝑃 βˆ’ 1) / 2) ↑ 𝑋) βˆ’ 1 )    &   πΏ = (β„€RHomβ€˜π‘Œ)    &   (πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ ((𝐴↑((𝑃 βˆ’ 1) / 2)) mod 𝑃) = (1 mod 𝑃))    β‡’   (πœ‘ β†’ ((π‘‚β€˜π‘‡)β€˜(πΏβ€˜π΄)) = (0gβ€˜π‘Œ))
 
Theoremlgsqrlem2 26617* Lemma for lgsqr 26621. (Contributed by Mario Carneiro, 15-Jun-2015.)
π‘Œ = (β„€/nβ„€β€˜π‘ƒ)    &   π‘† = (Poly1β€˜π‘Œ)    &   π΅ = (Baseβ€˜π‘†)    &   π· = ( deg1 β€˜π‘Œ)    &   π‘‚ = (eval1β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   π‘‹ = (var1β€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘†)    &    1 = (1rβ€˜π‘†)    &   π‘‡ = ((((𝑃 βˆ’ 1) / 2) ↑ 𝑋) βˆ’ 1 )    &   πΏ = (β„€RHomβ€˜π‘Œ)    &   (πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   πΊ = (𝑦 ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ (πΏβ€˜(𝑦↑2)))    β‡’   (πœ‘ β†’ 𝐺:(1...((𝑃 βˆ’ 1) / 2))–1-1β†’(β—‘(π‘‚β€˜π‘‡) β€œ {(0gβ€˜π‘Œ)}))
 
Theoremlgsqrlem3 26618* Lemma for lgsqr 26621. (Contributed by Mario Carneiro, 15-Jun-2015.)
π‘Œ = (β„€/nβ„€β€˜π‘ƒ)    &   π‘† = (Poly1β€˜π‘Œ)    &   π΅ = (Baseβ€˜π‘†)    &   π· = ( deg1 β€˜π‘Œ)    &   π‘‚ = (eval1β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   π‘‹ = (var1β€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘†)    &    1 = (1rβ€˜π‘†)    &   π‘‡ = ((((𝑃 βˆ’ 1) / 2) ↑ 𝑋) βˆ’ 1 )    &   πΏ = (β„€RHomβ€˜π‘Œ)    &   (πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   πΊ = (𝑦 ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ (πΏβ€˜(𝑦↑2)))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ (𝐴 /L 𝑃) = 1)    β‡’   (πœ‘ β†’ (πΏβ€˜π΄) ∈ (β—‘(π‘‚β€˜π‘‡) β€œ {(0gβ€˜π‘Œ)}))
 
Theoremlgsqrlem4 26619* Lemma for lgsqr 26621. (Contributed by Mario Carneiro, 15-Jun-2015.)
π‘Œ = (β„€/nβ„€β€˜π‘ƒ)    &   π‘† = (Poly1β€˜π‘Œ)    &   π΅ = (Baseβ€˜π‘†)    &   π· = ( deg1 β€˜π‘Œ)    &   π‘‚ = (eval1β€˜π‘Œ)    &    ↑ = (.gβ€˜(mulGrpβ€˜π‘†))    &   π‘‹ = (var1β€˜π‘Œ)    &    βˆ’ = (-gβ€˜π‘†)    &    1 = (1rβ€˜π‘†)    &   π‘‡ = ((((𝑃 βˆ’ 1) / 2) ↑ 𝑋) βˆ’ 1 )    &   πΏ = (β„€RHomβ€˜π‘Œ)    &   (πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   πΊ = (𝑦 ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ (πΏβ€˜(𝑦↑2)))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ (𝐴 /L 𝑃) = 1)    β‡’   (πœ‘ β†’ βˆƒπ‘₯ ∈ β„€ 𝑃 βˆ₯ ((π‘₯↑2) βˆ’ 𝐴))
 
Theoremlgsqrlem5 26620* Lemma for lgsqr 26621. (Contributed by Mario Carneiro, 15-Jun-2015.)
((𝐴 ∈ β„€ ∧ 𝑃 ∈ (β„™ βˆ– {2}) ∧ (𝐴 /L 𝑃) = 1) β†’ βˆƒπ‘₯ ∈ β„€ 𝑃 βˆ₯ ((π‘₯↑2) βˆ’ 𝐴))
 
Theoremlgsqr 26621* The Legendre symbol for odd primes is 1 iff the number is not a multiple of the prime (in which case it is 0, see lgsne0 26605) and the number is a quadratic residue mod 𝑃 (it is -1 for nonresidues by the process of elimination from lgsabs1 26606). Given our definition of the Legendre symbol, this theorem is equivalent to Euler's criterion. (Contributed by Mario Carneiro, 15-Jun-2015.)
((𝐴 ∈ β„€ ∧ 𝑃 ∈ (β„™ βˆ– {2})) β†’ ((𝐴 /L 𝑃) = 1 ↔ (Β¬ 𝑃 βˆ₯ 𝐴 ∧ βˆƒπ‘₯ ∈ β„€ 𝑃 βˆ₯ ((π‘₯↑2) βˆ’ 𝐴))))
 
Theoremlgsqrmod 26622* If the Legendre symbol of an integer for an odd prime is 1, then the number is a quadratic residue mod 𝑃. (Contributed by AV, 20-Aug-2021.)
((𝐴 ∈ β„€ ∧ 𝑃 ∈ (β„™ βˆ– {2})) β†’ ((𝐴 /L 𝑃) = 1 β†’ βˆƒπ‘₯ ∈ β„€ ((π‘₯↑2) mod 𝑃) = (𝐴 mod 𝑃)))
 
Theoremlgsqrmodndvds 26623* If the Legendre symbol of an integer 𝐴 for an odd prime is 1, then the number is a quadratic residue mod 𝑃 with a solution π‘₯ of the congruence (π‘₯↑2)≑𝐴 (mod 𝑃) which is not divisible by the prime. (Contributed by AV, 20-Aug-2021.) (Proof shortened by AV, 18-Mar-2022.)
((𝐴 ∈ β„€ ∧ 𝑃 ∈ (β„™ βˆ– {2})) β†’ ((𝐴 /L 𝑃) = 1 β†’ βˆƒπ‘₯ ∈ β„€ (((π‘₯↑2) mod 𝑃) = (𝐴 mod 𝑃) ∧ Β¬ 𝑃 βˆ₯ π‘₯)))
 
Theoremlgsdchrval 26624* The Legendre symbol function 𝑋(π‘š) = (π‘š /L 𝑁), where 𝑁 is an odd positive number, is a Dirichlet character modulo 𝑁. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChrβ€˜π‘)    &   π‘ = (β„€/nβ„€β€˜π‘)    &   π· = (Baseβ€˜πΊ)    &   π΅ = (Baseβ€˜π‘)    &   πΏ = (β„€RHomβ€˜π‘)    &   π‘‹ = (𝑦 ∈ 𝐡 ↦ (β„©β„Žβˆƒπ‘š ∈ β„€ (𝑦 = (πΏβ€˜π‘š) ∧ β„Ž = (π‘š /L 𝑁))))    β‡’   (((𝑁 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑁) ∧ 𝐴 ∈ β„€) β†’ (π‘‹β€˜(πΏβ€˜π΄)) = (𝐴 /L 𝑁))
 
Theoremlgsdchr 26625* The Legendre symbol function 𝑋(π‘š) = (π‘š /L 𝑁), where 𝑁 is an odd positive number, is a real Dirichlet character modulo 𝑁. (Contributed by Mario Carneiro, 28-Apr-2016.)
𝐺 = (DChrβ€˜π‘)    &   π‘ = (β„€/nβ„€β€˜π‘)    &   π· = (Baseβ€˜πΊ)    &   π΅ = (Baseβ€˜π‘)    &   πΏ = (β„€RHomβ€˜π‘)    &   π‘‹ = (𝑦 ∈ 𝐡 ↦ (β„©β„Žβˆƒπ‘š ∈ β„€ (𝑦 = (πΏβ€˜π‘š) ∧ β„Ž = (π‘š /L 𝑁))))    β‡’   ((𝑁 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑁) β†’ (𝑋 ∈ 𝐷 ∧ 𝑋:π΅βŸΆβ„))
 
14.4.9  Gauss' Lemma

Gauss' Lemma is valid for any integer not dividing the given prime number. In the following, only the special case for 2 (not dividing any odd prime) is proven, see gausslemma2d 26644. The general case is still to prove.

 
Theoremgausslemma2dlem0a 26626 Auxiliary lemma 1 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    β‡’   (πœ‘ β†’ 𝑃 ∈ β„•)
 
Theoremgausslemma2dlem0b 26627 Auxiliary lemma 2 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    β‡’   (πœ‘ β†’ 𝐻 ∈ β„•)
 
Theoremgausslemma2dlem0c 26628 Auxiliary lemma 3 for gausslemma2d 26644. (Contributed by AV, 13-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    β‡’   (πœ‘ β†’ ((!β€˜π») gcd 𝑃) = 1)
 
Theoremgausslemma2dlem0d 26629 Auxiliary lemma 4 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    β‡’   (πœ‘ β†’ 𝑀 ∈ β„•0)
 
Theoremgausslemma2dlem0e 26630 Auxiliary lemma 5 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    β‡’   (πœ‘ β†’ (𝑀 Β· 2) < (𝑃 / 2))
 
Theoremgausslemma2dlem0f 26631 Auxiliary lemma 6 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π» = ((𝑃 βˆ’ 1) / 2)    β‡’   (πœ‘ β†’ (𝑀 + 1) ≀ 𝐻)
 
Theoremgausslemma2dlem0g 26632 Auxiliary lemma 7 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π» = ((𝑃 βˆ’ 1) / 2)    β‡’   (πœ‘ β†’ 𝑀 ≀ 𝐻)
 
Theoremgausslemma2dlem0h 26633 Auxiliary lemma 8 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘ = (𝐻 βˆ’ 𝑀)    β‡’   (πœ‘ β†’ 𝑁 ∈ β„•0)
 
Theoremgausslemma2dlem0i 26634 Auxiliary lemma 9 for gausslemma2d 26644. (Contributed by AV, 14-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘ = (𝐻 βˆ’ 𝑀)    β‡’   (πœ‘ β†’ (((2 /L 𝑃) mod 𝑃) = ((-1↑𝑁) mod 𝑃) β†’ (2 /L 𝑃) = (-1↑𝑁)))
 
Theoremgausslemma2dlem1a 26635* Lemma for gausslemma2dlem1 26636. (Contributed by AV, 1-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    β‡’   (πœ‘ β†’ ran 𝑅 = (1...𝐻))
 
Theoremgausslemma2dlem1 26636* Lemma 1 for gausslemma2d 26644. (Contributed by AV, 5-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    β‡’   (πœ‘ β†’ (!β€˜π») = βˆπ‘˜ ∈ (1...𝐻)(π‘…β€˜π‘˜))
 
Theoremgausslemma2dlem2 26637* Lemma 2 for gausslemma2d 26644. (Contributed by AV, 4-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    β‡’   (πœ‘ β†’ βˆ€π‘˜ ∈ (1...𝑀)(π‘…β€˜π‘˜) = (π‘˜ Β· 2))
 
Theoremgausslemma2dlem3 26638* Lemma 3 for gausslemma2d 26644. (Contributed by AV, 4-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    β‡’   (πœ‘ β†’ βˆ€π‘˜ ∈ ((𝑀 + 1)...𝐻)(π‘…β€˜π‘˜) = (𝑃 βˆ’ (π‘˜ Β· 2)))
 
Theoremgausslemma2dlem4 26639* Lemma 4 for gausslemma2d 26644. (Contributed by AV, 16-Jun-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    β‡’   (πœ‘ β†’ (!β€˜π») = (βˆπ‘˜ ∈ (1...𝑀)(π‘…β€˜π‘˜) Β· βˆπ‘˜ ∈ ((𝑀 + 1)...𝐻)(π‘…β€˜π‘˜)))
 
Theoremgausslemma2dlem5a 26640* Lemma for gausslemma2dlem5 26641. (Contributed by AV, 8-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    β‡’   (πœ‘ β†’ (βˆπ‘˜ ∈ ((𝑀 + 1)...𝐻)(π‘…β€˜π‘˜) mod 𝑃) = (βˆπ‘˜ ∈ ((𝑀 + 1)...𝐻)(-1 Β· (π‘˜ Β· 2)) mod 𝑃))
 
Theoremgausslemma2dlem5 26641* Lemma 5 for gausslemma2d 26644. (Contributed by AV, 9-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π‘ = (𝐻 βˆ’ 𝑀)    β‡’   (πœ‘ β†’ (βˆπ‘˜ ∈ ((𝑀 + 1)...𝐻)(π‘…β€˜π‘˜) mod 𝑃) = (((-1↑𝑁) Β· βˆπ‘˜ ∈ ((𝑀 + 1)...𝐻)(π‘˜ Β· 2)) mod 𝑃))
 
Theoremgausslemma2dlem6 26642* Lemma 6 for gausslemma2d 26644. (Contributed by AV, 16-Jun-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π‘ = (𝐻 βˆ’ 𝑀)    β‡’   (πœ‘ β†’ ((!β€˜π») mod 𝑃) = ((((-1↑𝑁) Β· (2↑𝐻)) Β· (!β€˜π»)) mod 𝑃))
 
Theoremgausslemma2dlem7 26643* Lemma 7 for gausslemma2d 26644. (Contributed by AV, 13-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π‘ = (𝐻 βˆ’ 𝑀)    β‡’   (πœ‘ β†’ (((-1↑𝑁) Β· (2↑𝐻)) mod 𝑃) = 1)
 
Theoremgausslemma2d 26644* Gauss' Lemma (see also theorem 9.6 in [ApostolNT] p. 182) for integer 2: Let p be an odd prime. Let S = {2, 4, 6, ..., p - 1}. Let n denote the number of elements of S whose least positive residue modulo p is greater than p/2. Then ( 2 | p ) = (-1)^n. (Contributed by AV, 14-Jul-2021.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   π» = ((𝑃 βˆ’ 1) / 2)    &   π‘… = (π‘₯ ∈ (1...𝐻) ↦ if((π‘₯ Β· 2) < (𝑃 / 2), (π‘₯ Β· 2), (𝑃 βˆ’ (π‘₯ Β· 2))))    &   π‘€ = (βŒŠβ€˜(𝑃 / 4))    &   π‘ = (𝐻 βˆ’ 𝑀)    β‡’   (πœ‘ β†’ (2 /L 𝑃) = (-1↑𝑁))
 
14.4.10  Quadratic reciprocity
 
Theoremlgseisenlem1 26645* Lemma for lgseisen 26649. If 𝑅(𝑒) = (𝑄 Β· 𝑒) mod 𝑃 and 𝑀(𝑒) = (-1↑𝑅(𝑒)) Β· 𝑅(𝑒), then for any even 1 ≀ 𝑒 ≀ 𝑃 βˆ’ 1, 𝑀(𝑒) is also an even integer 1 ≀ 𝑀(𝑒) ≀ 𝑃 βˆ’ 1. To simplify these statements, we divide all the even numbers by 2, so that it becomes the statement that 𝑀(π‘₯ / 2) = (-1↑𝑅(π‘₯ / 2)) Β· 𝑅(π‘₯ / 2) / 2 is an integer between 1 and (𝑃 βˆ’ 1) / 2. (Contributed by Mario Carneiro, 17-Jun-2015.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘… = ((𝑄 Β· (2 Β· π‘₯)) mod 𝑃)    &   π‘€ = (π‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ ((((-1↑𝑅) Β· 𝑅) mod 𝑃) / 2))    β‡’   (πœ‘ β†’ 𝑀:(1...((𝑃 βˆ’ 1) / 2))⟢(1...((𝑃 βˆ’ 1) / 2)))
 
Theoremlgseisenlem2 26646* Lemma for lgseisen 26649. The function 𝑀 is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 17-Jun-2015.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘… = ((𝑄 Β· (2 Β· π‘₯)) mod 𝑃)    &   π‘€ = (π‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ ((((-1↑𝑅) Β· 𝑅) mod 𝑃) / 2))    &   π‘† = ((𝑄 Β· (2 Β· 𝑦)) mod 𝑃)    β‡’   (πœ‘ β†’ 𝑀:(1...((𝑃 βˆ’ 1) / 2))–1-1-ontoβ†’(1...((𝑃 βˆ’ 1) / 2)))
 
Theoremlgseisenlem3 26647* Lemma for lgseisen 26649. (Contributed by Mario Carneiro, 17-Jun-2015.) (Proof shortened by AV, 28-Jul-2019.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘… = ((𝑄 Β· (2 Β· π‘₯)) mod 𝑃)    &   π‘€ = (π‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ ((((-1↑𝑅) Β· 𝑅) mod 𝑃) / 2))    &   π‘† = ((𝑄 Β· (2 Β· 𝑦)) mod 𝑃)    &   π‘Œ = (β„€/nβ„€β€˜π‘ƒ)    &   πΊ = (mulGrpβ€˜π‘Œ)    &   πΏ = (β„€RHomβ€˜π‘Œ)    β‡’   (πœ‘ β†’ (𝐺 Ξ£g (π‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ (πΏβ€˜((-1↑𝑅) Β· 𝑄)))) = (1rβ€˜π‘Œ))
 
Theoremlgseisenlem4 26648* Lemma for lgseisen 26649. The function 𝑀 is an injection (and hence a bijection by the pigeonhole principle). (Contributed by Mario Carneiro, 18-Jun-2015.) (Proof shortened by AV, 15-Jun-2019.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘… = ((𝑄 Β· (2 Β· π‘₯)) mod 𝑃)    &   π‘€ = (π‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2)) ↦ ((((-1↑𝑅) Β· 𝑅) mod 𝑃) / 2))    &   π‘† = ((𝑄 Β· (2 Β· 𝑦)) mod 𝑃)    &   π‘Œ = (β„€/nβ„€β€˜π‘ƒ)    &   πΊ = (mulGrpβ€˜π‘Œ)    &   πΏ = (β„€RHomβ€˜π‘Œ)    β‡’   (πœ‘ β†’ ((𝑄↑((𝑃 βˆ’ 1) / 2)) mod 𝑃) = ((-1↑Σπ‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2))(βŒŠβ€˜((𝑄 / 𝑃) Β· (2 Β· π‘₯)))) mod 𝑃))
 
Theoremlgseisen 26649* Eisenstein's lemma, an expression for (𝑃 /L 𝑄) when 𝑃, 𝑄 are distinct odd primes. (Contributed by Mario Carneiro, 18-Jun-2015.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    β‡’   (πœ‘ β†’ (𝑄 /L 𝑃) = (-1↑Σπ‘₯ ∈ (1...((𝑃 βˆ’ 1) / 2))(βŒŠβ€˜((𝑄 / 𝑃) Β· (2 Β· π‘₯)))))
 
Theoremlgsquadlem1 26650* Lemma for lgsquad 26653. Count the members of 𝑆 with odd coordinates. (Contributed by Mario Carneiro, 19-Jun-2015.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘€ = ((𝑃 βˆ’ 1) / 2)    &   π‘ = ((𝑄 βˆ’ 1) / 2)    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 Β· 𝑃) < (π‘₯ Β· 𝑄))}    β‡’   (πœ‘ β†’ (-1↑Σ𝑒 ∈ (((βŒŠβ€˜(𝑀 / 2)) + 1)...𝑀)(βŒŠβ€˜((𝑄 / 𝑃) Β· (2 Β· 𝑒)))) = (-1↑(β™―β€˜{𝑧 ∈ 𝑆 ∣ Β¬ 2 βˆ₯ (1st β€˜π‘§)})))
 
Theoremlgsquadlem2 26651* Lemma for lgsquad 26653. Count the members of 𝑆 with even coordinates, and combine with lgsquadlem1 26650 to get the total count of lattice points in 𝑆 (up to parity). (Contributed by Mario Carneiro, 18-Jun-2015.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘€ = ((𝑃 βˆ’ 1) / 2)    &   π‘ = ((𝑄 βˆ’ 1) / 2)    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 Β· 𝑃) < (π‘₯ Β· 𝑄))}    β‡’   (πœ‘ β†’ (𝑄 /L 𝑃) = (-1↑(β™―β€˜π‘†)))
 
Theoremlgsquadlem3 26652* Lemma for lgsquad 26653. (Contributed by Mario Carneiro, 18-Jun-2015.)
(πœ‘ β†’ 𝑃 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑄 ∈ (β„™ βˆ– {2}))    &   (πœ‘ β†’ 𝑃 β‰  𝑄)    &   π‘€ = ((𝑃 βˆ’ 1) / 2)    &   π‘ = ((𝑄 βˆ’ 1) / 2)    &   π‘† = {⟨π‘₯, π‘¦βŸ© ∣ ((π‘₯ ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 Β· 𝑃) < (π‘₯ Β· 𝑄))}    β‡’   (πœ‘ β†’ ((𝑃 /L 𝑄) Β· (𝑄 /L 𝑃)) = (-1↑(𝑀 Β· 𝑁)))
 
Theoremlgsquad 26653 The Law of Quadratic Reciprocity, see also theorem 9.8 in [ApostolNT] p. 185. If 𝑃 and 𝑄 are distinct odd primes, then the product of the Legendre symbols (𝑃 /L 𝑄) and (𝑄 /L 𝑃) is the parity of ((𝑃 βˆ’ 1) / 2) Β· ((𝑄 βˆ’ 1) / 2). This uses Eisenstein's proof, which also has a nice geometric interpretation - see https://en.wikipedia.org/wiki/Proofs_of_quadratic_reciprocity. This is Metamath 100 proof #7. (Contributed by Mario Carneiro, 19-Jun-2015.)
((𝑃 ∈ (β„™ βˆ– {2}) ∧ 𝑄 ∈ (β„™ βˆ– {2}) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 /L 𝑄) Β· (𝑄 /L 𝑃)) = (-1↑(((𝑃 βˆ’ 1) / 2) Β· ((𝑄 βˆ’ 1) / 2))))
 
Theoremlgsquad2lem1 26654 Lemma for lgsquad2 26656. (Contributed by Mario Carneiro, 19-Jun-2015.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑀)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑁)    &   (πœ‘ β†’ (𝑀 gcd 𝑁) = 1)    &   (πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ (𝐴 Β· 𝐡) = 𝑀)    &   (πœ‘ β†’ ((𝐴 /L 𝑁) Β· (𝑁 /L 𝐴)) = (-1↑(((𝐴 βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))))    &   (πœ‘ β†’ ((𝐡 /L 𝑁) Β· (𝑁 /L 𝐡)) = (-1↑(((𝐡 βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))))    β‡’   (πœ‘ β†’ ((𝑀 /L 𝑁) Β· (𝑁 /L 𝑀)) = (-1↑(((𝑀 βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))))
 
Theoremlgsquad2lem2 26655* Lemma for lgsquad2 26656. (Contributed by Mario Carneiro, 19-Jun-2015.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑀)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑁)    &   (πœ‘ β†’ (𝑀 gcd 𝑁) = 1)    &   ((πœ‘ ∧ (π‘š ∈ (β„™ βˆ– {2}) ∧ (π‘š gcd 𝑁) = 1)) β†’ ((π‘š /L 𝑁) Β· (𝑁 /L π‘š)) = (-1↑(((π‘š βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))))    &   (πœ“ ↔ βˆ€π‘₯ ∈ (1...π‘˜)((π‘₯ gcd (2 Β· 𝑁)) = 1 β†’ ((π‘₯ /L 𝑁) Β· (𝑁 /L π‘₯)) = (-1↑(((π‘₯ βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2)))))    β‡’   (πœ‘ β†’ ((𝑀 /L 𝑁) Β· (𝑁 /L 𝑀)) = (-1↑(((𝑀 βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))))
 
Theoremlgsquad2 26656 Extend lgsquad 26653 to coprime odd integers (the domain of the Jacobi symbol). (Contributed by Mario Carneiro, 19-Jun-2015.)
(πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑀)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ Β¬ 2 βˆ₯ 𝑁)    &   (πœ‘ β†’ (𝑀 gcd 𝑁) = 1)    β‡’   (πœ‘ β†’ ((𝑀 /L 𝑁) Β· (𝑁 /L 𝑀)) = (-1↑(((𝑀 βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))))
 
Theoremlgsquad3 26657 Extend lgsquad2 26656 to integers which share a factor. (Contributed by Mario Carneiro, 19-Jun-2015.)
(((𝑀 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑀) ∧ (𝑁 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑁)) β†’ (𝑀 /L 𝑁) = ((-1↑(((𝑀 βˆ’ 1) / 2) Β· ((𝑁 βˆ’ 1) / 2))) Β· (𝑁 /L 𝑀)))
 
Theoremm1lgs 26658 The first supplement to the law of quadratic reciprocity. Negative one is a square mod an odd prime 𝑃 iff 𝑃≑1 (mod 4). See first case of theorem 9.4 in [ApostolNT] p. 181. (Contributed by Mario Carneiro, 19-Jun-2015.)
(𝑃 ∈ (β„™ βˆ– {2}) β†’ ((-1 /L 𝑃) = 1 ↔ (𝑃 mod 4) = 1))
 
Theorem2lgslem1a1 26659* Lemma 1 for 2lgslem1a 26661. (Contributed by AV, 16-Jun-2021.)
((𝑃 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑃) β†’ βˆ€π‘– ∈ (1...((𝑃 βˆ’ 1) / 2))(𝑖 Β· 2) = ((𝑖 Β· 2) mod 𝑃))
 
Theorem2lgslem1a2 26660 Lemma 2 for 2lgslem1a 26661. (Contributed by AV, 18-Jun-2021.)
((𝑁 ∈ β„€ ∧ 𝐼 ∈ β„€) β†’ ((βŒŠβ€˜(𝑁 / 4)) < 𝐼 ↔ (𝑁 / 2) < (𝐼 Β· 2)))
 
Theorem2lgslem1a 26661* Lemma 1 for 2lgslem1 26664. (Contributed by AV, 18-Jun-2021.)
((𝑃 ∈ β„™ ∧ Β¬ 2 βˆ₯ 𝑃) β†’ {π‘₯ ∈ β„€ ∣ βˆƒπ‘– ∈ (1...((𝑃 βˆ’ 1) / 2))(π‘₯ = (𝑖 Β· 2) ∧ (𝑃 / 2) < (π‘₯ mod 𝑃))} = {π‘₯ ∈ β„€ ∣ βˆƒπ‘– ∈ (((βŒŠβ€˜(𝑃 / 4)) + 1)...((𝑃 βˆ’ 1) / 2))π‘₯ = (𝑖 Β· 2)})
 
Theorem2lgslem1b 26662* Lemma 2 for 2lgslem1 26664. (Contributed by AV, 18-Jun-2021.)
𝐼 = (𝐴...𝐡)    &   πΉ = (𝑗 ∈ 𝐼 ↦ (𝑗 Β· 2))    β‡’   πΉ:𝐼–1-1-ontoβ†’{π‘₯ ∈ β„€ ∣ βˆƒπ‘– ∈ 𝐼 π‘₯ = (𝑖 Β· 2)}
 
Theorem2lgslem1c 26663 Lemma 3 for 2lgslem1 26664. (Contributed by AV, 19-Jun-2021.)
((𝑃 ∈ β„™ ∧ Β¬ 2 βˆ₯ 𝑃) β†’ (βŒŠβ€˜(𝑃 / 4)) ≀ ((𝑃 βˆ’ 1) / 2))
 
Theorem2lgslem1 26664* Lemma 1 for 2lgs 26677. (Contributed by AV, 19-Jun-2021.)
((𝑃 ∈ β„™ ∧ Β¬ 2 βˆ₯ 𝑃) β†’ (β™―β€˜{π‘₯ ∈ β„€ ∣ βˆƒπ‘– ∈ (1...((𝑃 βˆ’ 1) / 2))(π‘₯ = (𝑖 Β· 2) ∧ (𝑃 / 2) < (π‘₯ mod 𝑃))}) = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4))))
 
Theorem2lgslem2 26665 Lemma 2 for 2lgs 26677. (Contributed by AV, 20-Jun-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝑃 ∈ β„™ ∧ Β¬ 2 βˆ₯ 𝑃) β†’ 𝑁 ∈ β„€)
 
Theorem2lgslem3a 26666 Lemma for 2lgslem3a1 26670. (Contributed by AV, 14-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝐾 ∈ β„•0 ∧ 𝑃 = ((8 Β· 𝐾) + 1)) β†’ 𝑁 = (2 Β· 𝐾))
 
Theorem2lgslem3b 26667 Lemma for 2lgslem3b1 26671. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝐾 ∈ β„•0 ∧ 𝑃 = ((8 Β· 𝐾) + 3)) β†’ 𝑁 = ((2 Β· 𝐾) + 1))
 
Theorem2lgslem3c 26668 Lemma for 2lgslem3c1 26672. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝐾 ∈ β„•0 ∧ 𝑃 = ((8 Β· 𝐾) + 5)) β†’ 𝑁 = ((2 Β· 𝐾) + 1))
 
Theorem2lgslem3d 26669 Lemma for 2lgslem3d1 26673. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝐾 ∈ β„•0 ∧ 𝑃 = ((8 Β· 𝐾) + 7)) β†’ 𝑁 = ((2 Β· 𝐾) + 2))
 
Theorem2lgslem3a1 26670 Lemma 1 for 2lgslem3 26674. (Contributed by AV, 15-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝑃 ∈ β„• ∧ (𝑃 mod 8) = 1) β†’ (𝑁 mod 2) = 0)
 
Theorem2lgslem3b1 26671 Lemma 2 for 2lgslem3 26674. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝑃 ∈ β„• ∧ (𝑃 mod 8) = 3) β†’ (𝑁 mod 2) = 1)
 
Theorem2lgslem3c1 26672 Lemma 3 for 2lgslem3 26674. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝑃 ∈ β„• ∧ (𝑃 mod 8) = 5) β†’ (𝑁 mod 2) = 1)
 
Theorem2lgslem3d1 26673 Lemma 4 for 2lgslem3 26674. (Contributed by AV, 15-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝑃 ∈ β„• ∧ (𝑃 mod 8) = 7) β†’ (𝑁 mod 2) = 0)
 
Theorem2lgslem3 26674 Lemma 3 for 2lgs 26677. (Contributed by AV, 16-Jul-2021.)
𝑁 = (((𝑃 βˆ’ 1) / 2) βˆ’ (βŒŠβ€˜(𝑃 / 4)))    β‡’   ((𝑃 ∈ β„• ∧ Β¬ 2 βˆ₯ 𝑃) β†’ (𝑁 mod 2) = if((𝑃 mod 8) ∈ {1, 7}, 0, 1))
 
Theorem2lgs2 26675 The Legendre symbol for 2 at 2 is 0. (Contributed by AV, 20-Jun-2021.)
(2 /L 2) = 0
 
Theorem2lgslem4 26676 Lemma 4 for 2lgs 26677: special case of 2lgs 26677 for 𝑃 = 2. (Contributed by AV, 20-Jun-2021.)
((2 /L 2) = 1 ↔ (2 mod 8) ∈ {1, 7})
 
Theorem2lgs 26677 The second supplement to the law of quadratic reciprocity (for the Legendre symbol extended to arbitrary primes as second argument). Two is a square modulo a prime 𝑃 iff 𝑃≑±1 (mod 8), see first case of theorem 9.5 in [ApostolNT] p. 181. This theorem justifies our definition of (𝑁 /L 2) (lgs2 26584) to some degree, by demanding that reciprocity extend to the case 𝑄 = 2. (Proposed by Mario Carneiro, 19-Jun-2015.) (Contributed by AV, 16-Jul-2021.)
(𝑃 ∈ β„™ β†’ ((2 /L 𝑃) = 1 ↔ (𝑃 mod 8) ∈ {1, 7}))
 
Theorem2lgsoddprmlem1 26678 Lemma 1 for 2lgsoddprm 26686. (Contributed by AV, 19-Jul-2021.)
((𝐴 ∈ β„€ ∧ 𝐡 ∈ β„€ ∧ 𝑁 = ((8 Β· 𝐴) + 𝐡)) β†’ (((𝑁↑2) βˆ’ 1) / 8) = (((8 Β· (𝐴↑2)) + (2 Β· (𝐴 Β· 𝐡))) + (((𝐡↑2) βˆ’ 1) / 8)))
 
Theorem2lgsoddprmlem2 26679 Lemma 2 for 2lgsoddprm 26686. (Contributed by AV, 19-Jul-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁 ∧ 𝑅 = (𝑁 mod 8)) β†’ (2 βˆ₯ (((𝑁↑2) βˆ’ 1) / 8) ↔ 2 βˆ₯ (((𝑅↑2) βˆ’ 1) / 8)))
 
Theorem2lgsoddprmlem3a 26680 Lemma 1 for 2lgsoddprmlem3 26684. (Contributed by AV, 20-Jul-2021.)
(((1↑2) βˆ’ 1) / 8) = 0
 
Theorem2lgsoddprmlem3b 26681 Lemma 2 for 2lgsoddprmlem3 26684. (Contributed by AV, 20-Jul-2021.)
(((3↑2) βˆ’ 1) / 8) = 1
 
Theorem2lgsoddprmlem3c 26682 Lemma 3 for 2lgsoddprmlem3 26684. (Contributed by AV, 20-Jul-2021.)
(((5↑2) βˆ’ 1) / 8) = 3
 
Theorem2lgsoddprmlem3d 26683 Lemma 4 for 2lgsoddprmlem3 26684. (Contributed by AV, 20-Jul-2021.)
(((7↑2) βˆ’ 1) / 8) = (2 Β· 3)
 
Theorem2lgsoddprmlem3 26684 Lemma 3 for 2lgsoddprm 26686. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁 ∧ 𝑅 = (𝑁 mod 8)) β†’ (2 βˆ₯ (((𝑅↑2) βˆ’ 1) / 8) ↔ 𝑅 ∈ {1, 7}))
 
Theorem2lgsoddprmlem4 26685 Lemma 4 for 2lgsoddprm 26686. (Contributed by AV, 20-Jul-2021.)
((𝑁 ∈ β„€ ∧ Β¬ 2 βˆ₯ 𝑁) β†’ (2 βˆ₯ (((𝑁↑2) βˆ’ 1) / 8) ↔ (𝑁 mod 8) ∈ {1, 7}))
 
Theorem2lgsoddprm 26686 The second supplement to the law of quadratic reciprocity for odd primes (common representation, see theorem 9.5 in [ApostolNT] p. 181): The Legendre symbol for 2 at an odd prime is minus one to the power of the square of the odd prime minus one divided by eight ((2 /L 𝑃) = -1^(((P^2)-1)/8) ). (Contributed by AV, 20-Jul-2021.)
(𝑃 ∈ (β„™ βˆ– {2}) β†’ (2 /L 𝑃) = (-1↑(((𝑃↑2) βˆ’ 1) / 8)))
 
14.4.11  All primes 4n+1 are the sum of two squares
 
Theorem2sqlem1 26687* Lemma for 2sq 26700. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    β‡’   (𝐴 ∈ 𝑆 ↔ βˆƒπ‘₯ ∈ β„€[i] 𝐴 = ((absβ€˜π‘₯)↑2))
 
Theorem2sqlem2 26688* Lemma for 2sq 26700. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    β‡’   (𝐴 ∈ 𝑆 ↔ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ 𝐴 = ((π‘₯↑2) + (𝑦↑2)))
 
Theoremmul2sq 26689 Fibonacci's identity (actually due to Diophantus). The product of two sums of two squares is also a sum of two squares. We can take advantage of Gaussian integers here to trivialize the proof. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    β‡’   ((𝐴 ∈ 𝑆 ∧ 𝐡 ∈ 𝑆) β†’ (𝐴 Β· 𝐡) ∈ 𝑆)
 
Theorem2sqlem3 26690 Lemma for 2sqlem5 26692. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ β„™)    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„€)    &   (πœ‘ β†’ 𝐢 ∈ β„€)    &   (πœ‘ β†’ 𝐷 ∈ β„€)    &   (πœ‘ β†’ (𝑁 Β· 𝑃) = ((𝐴↑2) + (𝐡↑2)))    &   (πœ‘ β†’ 𝑃 = ((𝐢↑2) + (𝐷↑2)))    &   (πœ‘ β†’ 𝑃 βˆ₯ ((𝐢 Β· 𝐡) + (𝐴 Β· 𝐷)))    β‡’   (πœ‘ β†’ 𝑁 ∈ 𝑆)
 
Theorem2sqlem4 26691 Lemma for 2sqlem5 26692. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ β„™)    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„€)    &   (πœ‘ β†’ 𝐢 ∈ β„€)    &   (πœ‘ β†’ 𝐷 ∈ β„€)    &   (πœ‘ β†’ (𝑁 Β· 𝑃) = ((𝐴↑2) + (𝐡↑2)))    &   (πœ‘ β†’ 𝑃 = ((𝐢↑2) + (𝐷↑2)))    β‡’   (πœ‘ β†’ 𝑁 ∈ 𝑆)
 
Theorem2sqlem5 26692 Lemma for 2sq 26700. If a number that is a sum of two squares is divisible by a prime that is a sum of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑃 ∈ β„™)    &   (πœ‘ β†’ (𝑁 Β· 𝑃) ∈ 𝑆)    &   (πœ‘ β†’ 𝑃 ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝑁 ∈ 𝑆)
 
Theorem2sqlem6 26693* Lemma for 2sq 26700. If a number that is a sum of two squares is divisible by a number whose prime divisors are all sums of two squares, then the quotient is a sum of two squares. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   (πœ‘ β†’ 𝐴 ∈ β„•)    &   (πœ‘ β†’ 𝐡 ∈ β„•)    &   (πœ‘ β†’ βˆ€π‘ ∈ β„™ (𝑝 βˆ₯ 𝐡 β†’ 𝑝 ∈ 𝑆))    &   (πœ‘ β†’ (𝐴 Β· 𝐡) ∈ 𝑆)    β‡’   (πœ‘ β†’ 𝐴 ∈ 𝑆)
 
Theorem2sqlem7 26694* Lemma for 2sq 26700. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   π‘Œ = {𝑧 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ ((π‘₯ gcd 𝑦) = 1 ∧ 𝑧 = ((π‘₯↑2) + (𝑦↑2)))}    β‡’   π‘Œ βŠ† (𝑆 ∩ β„•)
 
Theorem2sqlem8a 26695* Lemma for 2sqlem8 26696. (Contributed by Mario Carneiro, 4-Jun-2016.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   π‘Œ = {𝑧 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ ((π‘₯ gcd 𝑦) = 1 ∧ 𝑧 = ((π‘₯↑2) + (𝑦↑2)))}    &   (πœ‘ β†’ βˆ€π‘ ∈ (1...(𝑀 βˆ’ 1))βˆ€π‘Ž ∈ π‘Œ (𝑏 βˆ₯ π‘Ž β†’ 𝑏 ∈ 𝑆))    &   (πœ‘ β†’ 𝑀 βˆ₯ 𝑁)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„€)    &   (πœ‘ β†’ (𝐴 gcd 𝐡) = 1)    &   (πœ‘ β†’ 𝑁 = ((𝐴↑2) + (𝐡↑2)))    &   πΆ = (((𝐴 + (𝑀 / 2)) mod 𝑀) βˆ’ (𝑀 / 2))    &   π· = (((𝐡 + (𝑀 / 2)) mod 𝑀) βˆ’ (𝑀 / 2))    β‡’   (πœ‘ β†’ (𝐢 gcd 𝐷) ∈ β„•)
 
Theorem2sqlem8 26696* Lemma for 2sq 26700. (Contributed by Mario Carneiro, 20-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   π‘Œ = {𝑧 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ ((π‘₯ gcd 𝑦) = 1 ∧ 𝑧 = ((π‘₯↑2) + (𝑦↑2)))}    &   (πœ‘ β†’ βˆ€π‘ ∈ (1...(𝑀 βˆ’ 1))βˆ€π‘Ž ∈ π‘Œ (𝑏 βˆ₯ π‘Ž β†’ 𝑏 ∈ 𝑆))    &   (πœ‘ β†’ 𝑀 βˆ₯ 𝑁)    &   (πœ‘ β†’ 𝑁 ∈ β„•)    &   (πœ‘ β†’ 𝑀 ∈ (β„€β‰₯β€˜2))    &   (πœ‘ β†’ 𝐴 ∈ β„€)    &   (πœ‘ β†’ 𝐡 ∈ β„€)    &   (πœ‘ β†’ (𝐴 gcd 𝐡) = 1)    &   (πœ‘ β†’ 𝑁 = ((𝐴↑2) + (𝐡↑2)))    &   πΆ = (((𝐴 + (𝑀 / 2)) mod 𝑀) βˆ’ (𝑀 / 2))    &   π· = (((𝐡 + (𝑀 / 2)) mod 𝑀) βˆ’ (𝑀 / 2))    &   πΈ = (𝐢 / (𝐢 gcd 𝐷))    &   πΉ = (𝐷 / (𝐢 gcd 𝐷))    β‡’   (πœ‘ β†’ 𝑀 ∈ 𝑆)
 
Theorem2sqlem9 26697* Lemma for 2sq 26700. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   π‘Œ = {𝑧 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ ((π‘₯ gcd 𝑦) = 1 ∧ 𝑧 = ((π‘₯↑2) + (𝑦↑2)))}    &   (πœ‘ β†’ βˆ€π‘ ∈ (1...(𝑀 βˆ’ 1))βˆ€π‘Ž ∈ π‘Œ (𝑏 βˆ₯ π‘Ž β†’ 𝑏 ∈ 𝑆))    &   (πœ‘ β†’ 𝑀 βˆ₯ 𝑁)    &   (πœ‘ β†’ 𝑀 ∈ β„•)    &   (πœ‘ β†’ 𝑁 ∈ π‘Œ)    β‡’   (πœ‘ β†’ 𝑀 ∈ 𝑆)
 
Theorem2sqlem10 26698* Lemma for 2sq 26700. Every factor of a "proper" sum of two squares (where the summands are coprime) is a sum of two squares. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   π‘Œ = {𝑧 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ ((π‘₯ gcd 𝑦) = 1 ∧ 𝑧 = ((π‘₯↑2) + (𝑦↑2)))}    β‡’   ((𝐴 ∈ π‘Œ ∧ 𝐡 ∈ β„• ∧ 𝐡 βˆ₯ 𝐴) β†’ 𝐡 ∈ 𝑆)
 
Theorem2sqlem11 26699* Lemma for 2sq 26700. (Contributed by Mario Carneiro, 19-Jun-2015.)
𝑆 = ran (𝑀 ∈ β„€[i] ↦ ((absβ€˜π‘€)↑2))    &   π‘Œ = {𝑧 ∣ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ ((π‘₯ gcd 𝑦) = 1 ∧ 𝑧 = ((π‘₯↑2) + (𝑦↑2)))}    β‡’   ((𝑃 ∈ β„™ ∧ (𝑃 mod 4) = 1) β†’ 𝑃 ∈ 𝑆)
 
Theorem2sq 26700* All primes of the form 4π‘˜ + 1 are sums of two squares. This is Metamath 100 proof #20. (Contributed by Mario Carneiro, 20-Jun-2015.)
((𝑃 ∈ β„™ ∧ (𝑃 mod 4) = 1) β†’ βˆƒπ‘₯ ∈ β„€ βˆƒπ‘¦ ∈ β„€ 𝑃 = ((π‘₯↑2) + (𝑦↑2)))
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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 46901-46948
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