P. 450 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44901-45000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	257prm 44901	257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ ;;257 ∈ ℙ
Theorem	fmtno3prm 44902	The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ (FermatNo‘3) ∈ ℙ
Theorem	odz2prm2pw 44903	Any power of two is coprime to any prime not being two. (Contributed by AV, 25-Jul-2021.)
⊢ (((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2})) ∧ (((2↑(2↑𝑁)) mod 𝑃) ≠ 1 ∧ ((2↑(2↑(𝑁 + 1))) mod 𝑃) = 1)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
Theorem	fmtnoprmfac1lem 44904	Lemma for fmtnoprmfac1 44905: The order of 2 modulo a prime that divides the n-th Fermat number is 2^(n+1). (Contributed by AV, 25-Jul-2021.) (Proof shortened by AV, 18-Mar-2022.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((odℤ‘𝑃)‘2) = (2↑(𝑁 + 1)))
Theorem	fmtnoprmfac1 44905*	Divisor of Fermat number (special form of Euler's result, see fmtnofac1 44910): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+1)+1 where k is a positive integer. (Contributed by AV, 25-Jul-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
Theorem	fmtnoprmfac2lem1 44906	Lemma for fmtnoprmfac2 44907. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)
Theorem	fmtnoprmfac2 44907*	Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 44909): Let Fn be a Fermat number. Let p be a prime divisor of Fn. Then p is in the form: k*2^(n+2)+1 where k is a positive integer. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ 𝑃 = ((𝑘 · (2↑(𝑁 + 2))) + 1))
Theorem	fmtnofac2lem 44908*	Lemma for fmtnofac2 44909 (Induction step). (Contributed by AV, 30-Jul-2021.)
⊢ ((𝑦 ∈ (ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2)) → ((((𝑁 ∈ (ℤ≥‘2) ∧ 𝑦 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑦 = ((𝑘 · (2↑(𝑁 + 2))) + 1)) ∧ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑧 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑧 = ((𝑘 · (2↑(𝑁 + 2))) + 1))) → ((𝑁 ∈ (ℤ≥‘2) ∧ (𝑦 · 𝑧) ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 (𝑦 · 𝑧) = ((𝑘 · (2↑(𝑁 + 2))) + 1))))
Theorem	fmtnofac2 44909*	Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 44910: Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+2)+1 where k is a nonnegative integer. (Contributed by AV, 30-Jul-2021.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 2))) + 1))
Theorem	fmtnofac1 44910*	Divisor of Fermat number (Euler's Result), see ProofWiki "Divisor of Fermat Number/Euler's Result", 24-Jul-2021, https://proofwiki.org/wiki/Divisor_of_Fermat_Number/Euler's_Result): "Let Fn be a Fermat number. Let m be divisor of Fn. Then m is in the form: k*2^(n+1)+1 where k is a positive integer." Here, however, k must be a nonnegative integer, because k must be 0 to represent 1 (which is a divisor of Fn ). Historical Note: In 1747, Leonhard Paul Euler proved that a divisor of a Fermat number Fn is always in the form kx2^(n+1)+1. This was later refined to k*2^(n+2)+1 by François Édouard Anatole Lucas, see fmtnofac2 44909. (Contributed by AV, 30-Jul-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
Theorem	fmtno4sqrt 44911	The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
⊢ (⌊‘(√‘(FermatNo‘4))) = ;;256
Theorem	fmtno4prmfac 44912	If P was a (prime) factor of the fourth Fermat number less than the square root of the fourth Fermat number, it would be either 65 or 129 or 193. (Contributed by AV, 28-Jul-2021.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → (𝑃 = ;65 ∨ 𝑃 = ;;129 ∨ 𝑃 = ;;193))
Theorem	fmtno4prmfac193 44913	If P was a (prime) factor of the fourth Fermat number, it would be 193. (Contributed by AV, 28-Jul-2021.)
⊢ ((𝑃 ∈ ℙ ∧ 𝑃 ∥ (FermatNo‘4) ∧ 𝑃 ≤ (⌊‘(√‘(FermatNo‘4)))) → 𝑃 = ;;193)
Theorem	fmtno4nprmfac193 44914	193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
⊢ ¬ ;;193 ∥ (FermatNo‘4)
Theorem	fmtno4prm 44915	The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
⊢ (FermatNo‘4) ∈ ℙ
Theorem	65537prm 44916	65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
⊢ ;;;;65537 ∈ ℙ
Theorem	fmtnofz04prm 44917	The first five Fermat numbers are prime, see remark in [ApostolNT] p. 7. (Contributed by AV, 28-Jul-2021.)
⊢ (𝑁 ∈ (0...4) → (FermatNo‘𝑁) ∈ ℙ)
Theorem	fmtnole4prm 44918	The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)
Theorem	fmtno5faclem1 44919	Lemma 1 for fmtno5fac 44922. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668
Theorem	fmtno5faclem2 44920	Lemma 2 for fmtno5fac 44922. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502
Theorem	fmtno5faclem3 44921	Lemma 3 for fmtno5fac 44922. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;;;402025020 + ;;;;;;;26801668) = ;;;;;;;;428826688
Theorem	fmtno5fac 44922	The factorisation of the 5 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 22-Jul-2021.)
⊢ (FermatNo‘5) = (;;;;;;6700417 · ;;641)
Theorem	fmtno5nprm 44923	The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
⊢ (FermatNo‘5) ∉ ℙ
Theorem	prmdvdsfmtnof1lem1 44924*	Lemma 1 for prmdvdsfmtnof1 44927. (Contributed by AV, 3-Aug-2021.)
⊢ 𝐼 = inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐹}, ℝ, < )    &   ⊢ 𝐽 = inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝐺}, ℝ, < )    ⇒   ⊢ ((𝐹 ∈ (ℤ≥‘2) ∧ 𝐺 ∈ (ℤ≥‘2)) → (𝐼 = 𝐽 → (𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺)))
Theorem	prmdvdsfmtnof1lem2 44925	Lemma 2 for prmdvdsfmtnof1 44927. (Contributed by AV, 3-Aug-2021.)
⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
Theorem	prmdvdsfmtnof 44926*	The mapping of a Fermat number to its smallest prime factor is a function. (Contributed by AV, 4-Aug-2021.) (Proof shortened by II, 16-Feb-2023.)
⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ))    ⇒   ⊢ 𝐹:ran FermatNo⟶ℙ
Theorem	prmdvdsfmtnof1 44927*	The mapping of a Fermat number to its smallest prime factor is a one-to-one function. (Contributed by AV, 4-Aug-2021.)
⊢ 𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝 ∥ 𝑓}, ℝ, < ))    ⇒   ⊢ 𝐹:ran FermatNo–1-1→ℙ
Theorem	prminf2 44928	The set of prime numbers is infinite. The proof of this variant of prminf 16544 is based on Goldbach's theorem goldbachth 44887 (via prmdvdsfmtnof1 44927 and prmdvdsfmtnof1lem2 44925), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties 44925. (Contributed by AV, 4-Aug-2021.)
⊢ ℙ ∉ Fin
Theorem	2pwp1prm 44929*	For ((2↑𝑘) + 1) to be prime, 𝑘 must be a power of 2, see Wikipedia "Fermat number", section "Other theorms about Fermat numbers", https://en.wikipedia.org/wiki/Fermat_number, 5-Aug-2021. (Contributed by AV, 7-Aug-2021.)
⊢ ((𝐾 ∈ ℕ ∧ ((2↑𝐾) + 1) ∈ ℙ) → ∃𝑛 ∈ ℕ0 𝐾 = (2↑𝑛))
Theorem	2pwp1prmfmtno 44930*	Every prime number of the form ((2↑𝑘) + 1) must be a Fermat number. (Contributed by AV, 7-Aug-2021.)
⊢ ((𝐾 ∈ ℕ ∧ 𝑃 = ((2↑𝐾) + 1) ∧ 𝑃 ∈ ℙ) → ∃𝑛 ∈ ℕ0 𝑃 = (FermatNo‘𝑛))
20.41.12.2  Mersenne primes "In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form Mn = 2^n-1 for some integer n. They are named after Marin Mersenne ... If n is a composite number then so is 2^n-1. Therefore, an equivalent definition of the Mersenne primes is that they are the prime numbers of the form Mp = 2^p-1 for some prime p.", see Wikipedia "Mersenne prime", 16-Aug-2021, https://en.wikipedia.org/wiki/Mersenne_prime. See also definition in [ApostolNT] p. 4. This means that if Mn = 2^n-1 is prime, than n must be prime, too, see mersenne 26280. The reverse direction is not generally valid: If p is prime, then Mp = 2^p-1 needs not be prime, e.g. M11 = 2047 = 23 x 89, see m11nprm 44941. This is an example of sgprmdvdsmersenne 44944, stating that if p with p = 3 modulo 4 (here 11) and q=2p+1 (here 23) are prime, then q divides Mp. "In number theory, a prime number p is a Sophie Germain prime if 2p+1 is also prime. The number 2p+1 associated with a Sophie Germain prime is called a safe prime.", see Wikipedia "Safe and Sophie Germain primes", 21-Aug-2021, https://en.wikipedia.org/wiki/Safe_and_Sophie_Germain_primes 44944. Hence, 11 is a Sophie Germain prime and 2x11+1=23 is its associated safe prime. By sfprmdvdsmersenne 44943, it is shown that if a safe prime q is congruent to 7 modulo 8, then it is a divisor of the Mersenne number with its matching Sophie Germain prime as exponent. The main result of this section, however, is the formal proof of a theorem of S. Ligh and L. Neal in "A note on Mersenne numbers", see lighneal 44951.
Theorem	m2prm 44931	The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.)
⊢ ((2↑2) − 1) ∈ ℙ
Theorem	m3prm 44932	The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.)
⊢ ((2↑3) − 1) ∈ ℙ
Theorem	flsqrt 44933	A condition equivalent to the floor of a square root. (Contributed by AV, 17-Aug-2021.)
⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ 𝐵 ∈ ℕ0) → ((⌊‘(√‘𝐴)) = 𝐵 ↔ ((𝐵↑2) ≤ 𝐴 ∧ 𝐴 < ((𝐵 + 1)↑2))))
Theorem	flsqrt5 44934	The floor of the square root of a nonnegative number is 5 iff the number is between 25 and 35. (Contributed by AV, 17-Aug-2021.)
⊢ ((𝑋 ∈ ℝ ∧ 0 ≤ 𝑋) → ((;25 ≤ 𝑋 ∧ 𝑋 < ;36) ↔ (⌊‘(√‘𝑋)) = 5))
Theorem	3ndvds4 44935	3 does not divide 4. (Contributed by AV, 18-Aug-2021.)
⊢ ¬ 3 ∥ 4
Theorem	139prmALT 44936	139 is a prime number. In contrast to 139prm 16753, the proof of this theorem uses 3dvds2dec 15970 for checking the divisibility by 3. Although the proof using 3dvds2dec 15970 is longer (regarding size: 1849 characters compared with 1809 for 139prm 16753), the number of essential steps is smaller (301 compared with 327 for 139prm 16753). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ;;139 ∈ ℙ
Theorem	31prm 44937	31 is a prime number. In contrast to 37prm 16750, the proof of this theorem is not based on the "blanket" prmlem2 16749, but on isprm7 16341. Although the checks for non-divisibility by the primes 7 to 23 are not needed, the proof is much longer (regarding size) than the proof of 37prm 16750 (1810 characters compared with 1213 for 37prm 16750). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 16750). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.)
⊢ ;31 ∈ ℙ
Theorem	m5prm 44938	The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.)
⊢ ((2↑5) − 1) ∈ ℙ
Theorem	127prm 44939	127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.)
⊢ ;;127 ∈ ℙ
Theorem	m7prm 44940	The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.)
⊢ ((2↑7) − 1) ∈ ℙ
Theorem	m11nprm 44941	The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.)
⊢ ((2↑;11) − 1) = (;89 · ;23)
Theorem	mod42tp1mod8 44942	If a number is 3 modulo 4, twice the number plus 1 is 7 modulo 8. (Contributed by AV, 19-Aug-2021.)
⊢ ((𝑁 ∈ ℤ ∧ (𝑁 mod 4) = 3) → (((2 · 𝑁) + 1) mod 8) = 7)
Theorem	sfprmdvdsmersenne 44943	If 𝑄 is a safe prime (i.e. 𝑄 = ((2 · 𝑃) + 1) for a prime 𝑃) with 𝑄≡7 (mod 8), then 𝑄 divides the 𝑃-th Mersenne number MP. (Contributed by AV, 20-Aug-2021.)
⊢ ((𝑃 ∈ ℙ ∧ (𝑄 ∈ ℙ ∧ (𝑄 mod 8) = 7 ∧ 𝑄 = ((2 · 𝑃) + 1))) → 𝑄 ∥ ((2↑𝑃) − 1))
Theorem	sgprmdvdsmersenne 44944	If 𝑃 is a Sophie Germain prime (i.e. 𝑄 = ((2 · 𝑃) + 1) is also prime) with 𝑃≡3 (mod 4), then 𝑄 divides the 𝑃-th Mersenne number MP. (Contributed by AV, 20-Aug-2021.)
⊢ (((𝑃 ∈ ℙ ∧ (𝑃 mod 4) = 3) ∧ (𝑄 = ((2 · 𝑃) + 1) ∧ 𝑄 ∈ ℙ)) → 𝑄 ∥ ((2↑𝑃) − 1))
Theorem	lighneallem1 44945	Lemma 1 for lighneal 44951. (Contributed by AV, 11-Aug-2021.)
⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃↑𝑀))
Theorem	lighneallem2 44946	Lemma 2 for lighneal 44951. (Contributed by AV, 13-Aug-2021.)
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)
Theorem	lighneallem3 44947	Lemma 3 for lighneal 44951. (Contributed by AV, 11-Aug-2021.)
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)
Theorem	lighneallem4a 44948	Lemma 1 for lighneallem4 44950. (Contributed by AV, 16-Aug-2021.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘3) ∧ 𝑆 = (((𝐴↑𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆)
Theorem	lighneallem4b 44949*	Lemma 2 for lighneallem4 44950. (Contributed by AV, 16-Aug-2021.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴↑𝑘)) ∈ (ℤ≥‘2))
Theorem	lighneallem4 44950	Lemma 3 for lighneal 44951. (Contributed by AV, 16-Aug-2021.)
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)
Theorem	lighneal 44951	If a power of a prime 𝑃 (i.e. 𝑃↑𝑀) is of the form 2↑𝑁 − 1, then 𝑁 must be prime and 𝑀 must be 1. Generalization of mersenne 26280 (where 𝑀 = 1 is a prerequisite). Theorem of S. Ligh and L. Neal (1974) "A note on Mersenne mumbers", Mathematics Magazine, 47:4, 231-233. (Contributed by AV, 16-Aug-2021.)
⊢ (((𝑃 ∈ ℙ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → (𝑀 = 1 ∧ 𝑁 ∈ ℙ))
20.41.12.3  Proth's theorem
Theorem	modexp2m1d 44952	The square of an integer which is -1 modulo a number greater than 1 is 1 modulo the same modulus. (Contributed by AV, 5-Jul-2020.)
⊢ (𝜑 → 𝐴 ∈ ℤ)    &   ⊢ (𝜑 → 𝐸 ∈ ℝ+)    &   ⊢ (𝜑 → 1 < 𝐸)    &   ⊢ (𝜑 → (𝐴 mod 𝐸) = (-1 mod 𝐸))    ⇒   ⊢ (𝜑 → ((𝐴↑2) mod 𝐸) = 1)
Theorem	proththdlem 44953	Lemma for proththd 44954. (Contributed by AV, 4-Jul-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 = ((𝐾 · (2↑𝑁)) + 1))    ⇒   ⊢ (𝜑 → (𝑃 ∈ ℕ ∧ 1 < 𝑃 ∧ ((𝑃 − 1) / 2) ∈ ℕ))
Theorem	proththd 44954*	Proth's theorem (1878). If P is a Proth number, i.e. a number of the form k2^n+1 with k less than 2^n, and if there exists an integer x for which x^((P-1)/2) is -1 modulo P, then P is prime. Such a prime is called a Proth prime. Like Pocklington's theorem (see pockthg 16535), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)
⊢ (𝜑 → 𝑁 ∈ ℕ)    &   ⊢ (𝜑 → 𝐾 ∈ ℕ)    &   ⊢ (𝜑 → 𝑃 = ((𝐾 · (2↑𝑁)) + 1))    &   ⊢ (𝜑 → 𝐾 < (2↑𝑁))    &   ⊢ (𝜑 → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))    ⇒   ⊢ (𝜑 → 𝑃 ∈ ℙ)
Theorem	5tcu2e40 44955	5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
⊢ (5 · (2↑3)) = ;40
Theorem	3exp4mod41 44956	3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
⊢ ((3↑4) mod ;41) = (-1 mod ;41)
Theorem	41prothprmlem1 44957	Lemma 1 for 41prothprm 44959. (Contributed by AV, 4-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ ((𝑃 − 1) / 2) = ;20
Theorem	41prothprmlem2 44958	Lemma 2 for 41prothprm 44959. (Contributed by AV, 5-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)
Theorem	41prothprm 44959	41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ)
20.41.12.4  Solutions of quadratic equations
Theorem	quad1 44960*	A condition for a quadratic equation with complex coefficients to have (exactly) one complex solution. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → 𝐶 ∈ ℂ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ ℂ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
Theorem	requad01 44961*	A condition for a quadratic equation with real coefficients to have (at least) one real solution. (Contributed by AV, 23-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 0 ≤ 𝐷))
Theorem	requad1 44962*	A condition for a quadratic equation with real coefficients to have (exactly) one real solution. (Contributed by AV, 26-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃!𝑥 ∈ ℝ ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0 ↔ 𝐷 = 0))
Theorem	requad2 44963*	A condition for a quadratic equation with real coefficients to have (exactly) two different real solutions. (Contributed by AV, 28-Jan-2023.)
⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 𝐴 ≠ 0)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐶 ∈ ℝ)    &   ⊢ (𝜑 → 𝐷 = ((𝐵↑2) − (4 · (𝐴 · 𝐶))))    ⇒   ⊢ (𝜑 → (∃!𝑝 ∈ 𝒫 ℝ((♯‘𝑝) = 2 ∧ ∀𝑥 ∈ 𝑝 ((𝐴 · (𝑥↑2)) + ((𝐵 · 𝑥) + 𝐶)) = 0) ↔ 0 < 𝐷))
20.41.13  Even and odd numbers Even and odd numbers can be characterized in many different ways. In the following, the definition of even and odd numbers is based on the fact that dividing an even number (resp. an odd number increased by 1) by 2 is an integer, see df-even 44966 and df-odd 44967. Alternate definitions resp. characterizations are provided in dfeven2 44989, dfeven3 44998, dfeven4 44978 and in dfodd2 44976, dfodd3 44990, dfodd4 44999, dfodd5 45000, dfodd6 44977. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 44977 in opoeALTV 45023 and dfodd3 44990 in oddprmALTV 45027. Having a fixed definition for even and odd numbers, and alternate characterizations as theorems, advanced theorems about even and/or odd numbers can be expressed more explicitly, and the appropriate characterization can be chosen for their proof, which may become clearer and sometimes also shorter (see, for example, divgcdoddALTV 45022 and divgcdodd 16343).
20.41.13.1  Definitions and basic properties
Syntax	ceven 44964	Extend the definition of a class to include the set of even numbers.
class Even
Syntax	codd 44965	Extend the definition of a class to include the set of odd numbers.
class Odd
Definition	df-even 44966	Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
Definition	df-odd 44967	Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
Theorem	iseven 44968	The predicate "is an even number". An even number is an integer which is divisible by 2, i.e. the result of dividing the even integer by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (𝑍 / 2) ∈ ℤ))
Theorem	isodd 44969	The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd integer increased by 1 and then divided by 2 is still an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 + 1) / 2) ∈ ℤ))
Theorem	evenz 44970	An even number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Theorem	oddz 44971	An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)
Theorem	evendiv2z 44972	The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)
Theorem	oddp1div2z 44973	The result of dividing an odd number increased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Odd → ((𝑍 + 1) / 2) ∈ ℤ)
Theorem	oddm1div2z 44974	The result of dividing an odd number decreased by 1 and then divided by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Odd → ((𝑍 − 1) / 2) ∈ ℤ)
Theorem	isodd2 44975	The predicate "is an odd number". An odd number is an integer which is not divisible by 2, i.e. the result of dividing the odd number decreased by 1 and then divided by 2 is still an integer. (Contributed by AV, 15-Jun-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ((𝑍 − 1) / 2) ∈ ℤ))
Theorem	dfodd2 44976	Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}
Theorem	dfodd6 44977*	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}
Theorem	dfeven4 44978*	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}
Theorem	evenm1odd 44979	The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )
Theorem	evenp1odd 44980	The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )
Theorem	oddp1eveni 44981	The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )
Theorem	oddm1eveni 44982	The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 − 1) ∈ Even )
Theorem	evennodd 44983	An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → ¬ 𝑍 ∈ Odd )
Theorem	oddneven 44984	An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )
Theorem	enege 44985	The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Even → -𝐴 ∈ Even )
Theorem	onego 44986	The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd )
Theorem	m1expevenALTV 44987	Exponentiation of -1 by an even power. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 6-Jul-2020.)
⊢ (𝑁 ∈ Even → (-1↑𝑁) = 1)
Theorem	m1expoddALTV 44988	Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1)
20.41.13.2  Alternate definitions using the "divides" relation
Theorem	dfeven2 44989	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧}
Theorem	dfodd3 44990	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}
Theorem	iseven2 44991	The predicate "is an even number". An even number is an integer which is divisible by 2. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ 2 ∥ 𝑍))
Theorem	isodd3 44992	The predicate "is an odd number". An odd number is an integer which is not divisible by 2. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ ¬ 2 ∥ 𝑍))
Theorem	2dvdseven 44993	2 divides an even number. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Even → 2 ∥ 𝑍)
Theorem	m2even 44994	A multiple of 2 is an even number. (Contributed by AV, 5-Jun-2023.)
⊢ (𝑍 ∈ ℤ → (2 · 𝑍) ∈ Even )
Theorem	2ndvdsodd 44995	2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 2 ∥ 𝑍)
Theorem	2dvdsoddp1 44996	2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 + 1))
Theorem	2dvdsoddm1 44997	2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 − 1))
20.41.13.3  Alternate definitions using the "modulo" operation
Theorem	dfeven3 44998	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0}
Theorem	dfodd4 44999	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1}
Theorem	dfodd5 45000	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0}