P. 468 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 46701-46800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	fmtnorec3 46701*	The third recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 2-Aug-2021.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (FermatNo‘𝑁) = ((FermatNo‘(𝑁 − 1)) + ((2↑(2↑(𝑁 − 1))) · ∏𝑛 ∈ (0...(𝑁 − 2))(FermatNo‘𝑛))))
Theorem	fmtnorec4 46702	The fourth recurrence relation for Fermat numbers, see Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 31-Jul-2021.)
⊢ (𝑁 ∈ (ℤ≥‘2) → (FermatNo‘𝑁) = (((FermatNo‘(𝑁 − 1))↑2) − (2 · (((FermatNo‘(𝑁 − 2)) − 1)↑2))))
Theorem	fmtno2 46703	The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘2) = ;17
Theorem	fmtno3 46704	The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘3) = ;;257
Theorem	fmtno4 46705	The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘4) = ;;;;65537
Theorem	fmtno5lem1 46706	Lemma 1 for fmtno5 46710. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 6) = ;;;;;393216
Theorem	fmtno5lem2 46707	Lemma 2 for fmtno5 46710. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 5) = ;;;;;327680
Theorem	fmtno5lem3 46708	Lemma 3 for fmtno5 46710. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;65536 · 3) = ;;;;;196608
Theorem	fmtno5lem4 46709	Lemma 4 for fmtno5 46710. (Contributed by AV, 30-Jul-2021.)
⊢ (;;;;65536↑2) = ;;;;;;;;;4294967296
Theorem	fmtno5 46710	The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
⊢ (FermatNo‘5) = ;;;;;;;;;4294967297
Theorem	fmtno0prm 46711	The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘0) ∈ ℙ
Theorem	fmtno1prm 46712	The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘1) ∈ ℙ
Theorem	fmtno2prm 46713	The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
⊢ (FermatNo‘2) ∈ ℙ
Theorem	257prm 46714	257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ ;;257 ∈ ℙ
Theorem	fmtno3prm 46715	The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
⊢ (FermatNo‘3) ∈ ℙ
Theorem	odz2prm2pw 46716	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 46717	Lemma for fmtnoprmfac1 46718: 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 46718*	Divisor of Fermat number (special form of Euler's result, see fmtnofac1 46723): 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 46719	Lemma for fmtnoprmfac2 46720. (Contributed by AV, 26-Jul-2021.)
⊢ ((𝑁 ∈ (ℤ≥‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)
Theorem	fmtnoprmfac2 46720*	Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 46722): 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 46721*	Lemma for fmtnofac2 46722 (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 46722*	Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 46723: 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 46723*	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 46722. (Contributed by AV, 30-Jul-2021.)
⊢ ((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))
Theorem	fmtno4sqrt 46724	The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
⊢ (⌊‘(√‘(FermatNo‘4))) = ;;256
Theorem	fmtno4prmfac 46725	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 46726	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 46727	193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
⊢ ¬ ;;193 ∥ (FermatNo‘4)
Theorem	fmtno4prm 46728	The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
⊢ (FermatNo‘4) ∈ ℙ
Theorem	65537prm 46729	65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
⊢ ;;;;65537 ∈ ℙ
Theorem	fmtnofz04prm 46730	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 46731	The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)
Theorem	fmtno5faclem1 46732	Lemma 1 for fmtno5fac 46735. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;6700417 · 4) = ;;;;;;;26801668
Theorem	fmtno5faclem2 46733	Lemma 2 for fmtno5fac 46735. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;6700417 · 6) = ;;;;;;;40202502
Theorem	fmtno5faclem3 46734	Lemma 3 for fmtno5fac 46735. (Contributed by AV, 22-Jul-2021.)
⊢ (;;;;;;;;402025020 + ;;;;;;;26801668) = ;;;;;;;;428826688
Theorem	fmtno5fac 46735	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 46736	The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
⊢ (FermatNo‘5) ∉ ℙ
Theorem	prmdvdsfmtnof1lem1 46737*	Lemma 1 for prmdvdsfmtnof1 46740. (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 46738	Lemma 2 for prmdvdsfmtnof1 46740. (Contributed by AV, 3-Aug-2021.)
⊢ ((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼 ∥ 𝐹 ∧ 𝐼 ∥ 𝐺) → 𝐹 = 𝐺))
Theorem	prmdvdsfmtnof 46739*	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 46740*	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 46741	The set of prime numbers is infinite. The proof of this variant of prminf 16847 is based on Goldbach's theorem goldbachth 46700 (via prmdvdsfmtnof1 46740 and prmdvdsfmtnof1lem2 46738), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties 46738. (Contributed by AV, 4-Aug-2021.)
⊢ ℙ ∉ Fin
Theorem	2pwp1prm 46742*	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 46743*	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))
21.45.12.2  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 27076. 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 46754. This is an example of sgprmdvdsmersenne 46757, 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 46757. Hence, 11 is a Sophie Germain prime and 2x11+1=23 is its associated safe prime. By sfprmdvdsmersenne 46756, 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 46764.
Theorem	m2prm 46744	The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.)
⊢ ((2↑2) − 1) ∈ ℙ
Theorem	m3prm 46745	The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.)
⊢ ((2↑3) − 1) ∈ ℙ
Theorem	flsqrt 46746	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 46747	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 46748	3 does not divide 4. (Contributed by AV, 18-Aug-2021.)
⊢ ¬ 3 ∥ 4
Theorem	139prmALT 46749	139 is a prime number. In contrast to 139prm 17056, the proof of this theorem uses 3dvds2dec 16273 for checking the divisibility by 3. Although the proof using 3dvds2dec 16273 is longer (regarding size: 1849 characters compared with 1809 for 139prm 17056), the number of essential steps is smaller (301 compared with 327 for 139prm 17056). (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 46750	31 is a prime number. In contrast to 37prm 17053, the proof of this theorem is not based on the "blanket" prmlem2 17052, but on isprm7 16642. 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 17053 (1810 characters compared with 1213 for 37prm 17053). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 17053). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.)
⊢ ;31 ∈ ℙ
Theorem	m5prm 46751	The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.)
⊢ ((2↑5) − 1) ∈ ℙ
Theorem	127prm 46752	127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.)
⊢ ;;127 ∈ ℙ
Theorem	m7prm 46753	The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.)
⊢ ((2↑7) − 1) ∈ ℙ
Theorem	m11nprm 46754	The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.)
⊢ ((2↑;11) − 1) = (;89 · ;23)
Theorem	mod42tp1mod8 46755	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 46756	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 46757	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 46758	Lemma 1 for lighneal 46764. (Contributed by AV, 11-Aug-2021.)
⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃↑𝑀))
Theorem	lighneallem2 46759	Lemma 2 for lighneal 46764. (Contributed by AV, 13-Aug-2021.)
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1)
Theorem	lighneallem3 46760	Lemma 3 for lighneal 46764. (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 46761	Lemma 1 for lighneallem4 46763. (Contributed by AV, 16-Aug-2021.)
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘3) ∧ 𝑆 = (((𝐴↑𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆)
Theorem	lighneallem4b 46762*	Lemma 2 for lighneallem4 46763. (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 46763	Lemma 3 for lighneal 46764. (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 46764	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 27076 (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))
21.45.12.3  Proth's theorem
Theorem	modexp2m1d 46765	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 46766	Lemma for proththd 46767. (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 46767*	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 16838), 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 46768	5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
⊢ (5 · (2↑3)) = ;40
Theorem	3exp4mod41 46769	3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
⊢ ((3↑4) mod ;41) = (-1 mod ;41)
Theorem	41prothprmlem1 46770	Lemma 1 for 41prothprm 46772. (Contributed by AV, 4-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ ((𝑃 − 1) / 2) = ;20
Theorem	41prothprmlem2 46771	Lemma 2 for 41prothprm 46772. (Contributed by AV, 5-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)
Theorem	41prothprm 46772	41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
⊢ 𝑃 = ;41    ⇒   ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ)
21.45.12.4  Solutions of quadratic equations
Theorem	quad1 46773*	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 46774*	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 46775*	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 46776*	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))
21.45.13  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 46779 and df-odd 46780. Alternate definitions resp. characterizations are provided in dfeven2 46802, dfeven3 46811, dfeven4 46791 and in dfodd2 46789, dfodd3 46803, dfodd4 46812, dfodd5 46813, dfodd6 46790. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 46790 in opoeALTV 46836 and dfodd3 46803 in oddprmALTV 46840. 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 46835 and divgcdodd 16644).
21.45.13.1  Definitions and basic properties
Syntax	ceven 46777	Extend the definition of a class to include the set of even numbers.
class Even
Syntax	codd 46778	Extend the definition of a class to include the set of odd numbers.
class Odd
Definition	df-even 46779	Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}
Definition	df-odd 46780	Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}
Theorem	iseven 46781	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 46782	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 46783	An even number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ)
Theorem	oddz 46784	An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ)
Theorem	evendiv2z 46785	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 46786	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 46787	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 46788	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 46789	Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}
Theorem	dfodd6 46790*	Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}
Theorem	dfeven4 46791*	Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}
Theorem	evenm1odd 46792	The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )
Theorem	evenp1odd 46793	The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )
Theorem	oddp1eveni 46794	The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )
Theorem	oddm1eveni 46795	The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
⊢ (𝑍 ∈ Odd → (𝑍 − 1) ∈ Even )
Theorem	evennodd 46796	An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Even → ¬ 𝑍 ∈ Odd )
Theorem	oddneven 46797	An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
⊢ (𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )
Theorem	enege 46798	The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Even → -𝐴 ∈ Even )
Theorem	onego 46799	The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd )
Theorem	m1expevenALTV 46800	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)