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Theorem List for Metamath Proof Explorer - 41801-41900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfmtnoprmfac1lem 41801 Lemma for fmtnoprmfac1 41802: 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)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))

Theoremfmtnoprmfac1 41802* Divisor of Fermat number (special form of Euler's result, see fmtnofac1 41807): 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))

Theoremfmtnoprmfac2lem1 41803 Lemma for fmtnoprmfac2 41804. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)

Theoremfmtnoprmfac2 41804* Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 41806): 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))

Theoremfmtnofac2lem 41805* Lemma for fmtnofac2 41806 (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))))

Theoremfmtnofac2 41806* Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 41807: 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))

Theoremfmtnofac1 41807* 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 41806. (Contributed by AV, 30-Jul-2021.)

((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑀 ∥ (FermatNo‘𝑁)) → ∃𝑘 ∈ ℕ0 𝑀 = ((𝑘 · (2↑(𝑁 + 1))) + 1))

Theoremfmtno4sqrt 41808 The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
(⌊‘(√‘(FermatNo‘4))) = 256

Theoremfmtno4prmfac 41809 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))

Theoremfmtno4prmfac193 41810 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)

Theoremfmtno4nprmfac193 41811 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
¬ 193 ∥ (FermatNo‘4)

Theoremfmtno4prm 41812 The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
(FermatNo‘4) ∈ ℙ

Theorem65537prm 41813 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
65537 ∈ ℙ

Theoremfmtnofz04prm 41814 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)

Theoremfmtnole4prm 41815 The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ0𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)

Theoremfmtno5faclem1 41816 Lemma 1 for fmtno5fac 41819. (Contributed by AV, 22-Jul-2021.)
(6700417 · 4) = 26801668

Theoremfmtno5faclem2 41817 Lemma 2 for fmtno5fac 41819. (Contributed by AV, 22-Jul-2021.)
(6700417 · 6) = 40202502

Theoremfmtno5faclem3 41818 Lemma 3 for fmtno5fac 41819. (Contributed by AV, 22-Jul-2021.)
(402025020 + 26801668) = 428826688

Theoremfmtno5fac 41819 The factorisation of the 5 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 22-Jul-2021.)
(FermatNo‘5) = (6700417 · 641)

Theoremfmtno5nprm 41820 The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
(FermatNo‘5) ∉ ℙ

Theoremprmdvdsfmtnof1lem1 41821* Lemma 1 for prmdvdsfmtnof1 41824. (Contributed by AV, 3-Aug-2021.)
𝐼 = inf({𝑝 ∈ ℙ ∣ 𝑝𝐹}, ℝ, < )    &   𝐽 = inf({𝑝 ∈ ℙ ∣ 𝑝𝐺}, ℝ, < )       ((𝐹 ∈ (ℤ‘2) ∧ 𝐺 ∈ (ℤ‘2)) → (𝐼 = 𝐽 → (𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺)))

Theoremprmdvdsfmtnof1lem2 41822 Lemma 2 for prmdvdsfmtnof1 41824. (Contributed by AV, 3-Aug-2021.)
((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))

Theoremprmdvdsfmtnof 41823* The mapping of a Fermat number to its smallest prime factor is a function. (Contributed by AV, 4-Aug-2021.)
𝐹 = (𝑓 ∈ ran FermatNo ↦ inf({𝑝 ∈ ℙ ∣ 𝑝𝑓}, ℝ, < ))       𝐹:ran FermatNo⟶ℙ

Theoremprmdvdsfmtnof1 41824* 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 ∣ 𝑝𝑓}, ℝ, < ))       𝐹:ran FermatNo–1-1→ℙ

Theoremprminf2 41825 The set of prime numbers is infinite. The proof of this variant of prminf 15666 is based on Goldbach's theorem goldbachth 41784 (via prmdvdsfmtnof1 41824 and prmdvdsfmtnof1lem2 41822), see Wikipedia "Fermat number", 4-Aug-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 4-Aug-2021.)
ℙ ∉ Fin

Theorempwdif 41826* The difference of two numbers to the same power is the difference of the two numbers multiplied with a finite sum. Generalization of subsq 13012. See Wikipedia "Fermat number", section "Other theorems about Fermat numbers", https://en.wikipedia.org/wiki/Fermat_number, 5-Aug-2021. (Contributed by AV, 6-Aug-2021.) (Revised by AV, 19-Aug-2021.)
((𝑁 ∈ ℕ0𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴𝑁) − (𝐵𝑁)) = ((𝐴𝐵) · Σ𝑘 ∈ (0..^𝑁)((𝐴𝑘) · (𝐵↑((𝑁𝑘) − 1)))))

Theorempwm1geoserALT 41827* The n-th power of a number decreased by 1 expressed by the finite geometric series 1 + 𝐴↑1 + 𝐴↑2 +... + 𝐴↑(𝑁 − 1). This alternate proof of pwm1geoser 14644 is not based on geoser 14643, but on pwdif 41826 and therefore shorter than the original proof. (Contributed by AV, 19-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝑁 ∈ ℕ0)       (𝜑 → ((𝐴𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴𝑘)))

Theorem2pwp1prm 41828* For every prime number of the form ((2↑𝑘) + 1) 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))

Theorem2pwp1prmfmtno 41829* 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))

20.35.7.2  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 24997. 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 41843. This is an example of sgprmdvdsmersenne 41846, 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. Hence, 11 is a Sophie Germain prime and 2x11+1=23 is its associated safe prime. By sfprmdvdsmersenne 41845, 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 41853.

Theoremm2prm 41830 The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑2) − 1) ∈ ℙ

Theoremm3prm 41831 The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑3) − 1) ∈ ℙ

Theorem2exp5 41832 Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
(2↑5) = 32

Theoremflsqrt 41833 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) ≤ 𝐴𝐴 < ((𝐵 + 1)↑2))))

Theoremflsqrt5 41834 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 ≤ 𝑋𝑋 < 36) ↔ (⌊‘(√‘𝑋)) = 5))

Theorem3ndvds4 41835 3 does not divide 4. (Contributed by AV, 18-Aug-2021.)
¬ 3 ∥ 4

Theorem139prmALT 41836 139 is a prime number. In contrast to 139prm 15878, the proof of this theorem uses 3dvds2dec 15103 for checking the divisibility by 3. Although the proof using 3dvds2dec 15103 is longer (regarding size: 1849 characters compared with 1809 for 139prm 15878), the number of essential steps is smaller (301 compared with 327 for 139prm 15878). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
139 ∈ ℙ

Theorem31prm 41837 31 is a prime number. In contrast to 37prm 15875, the proof of this theorem is not based on the "blanket" prmlem2 15874, but on isprm7 15467. 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 15875 (1810 characters compared with 1213 for 37prm 15875). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 15875). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.)
31 ∈ ℙ

Theoremm5prm 41838 The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.)
((2↑5) − 1) ∈ ℙ

Theorem2exp7 41839 Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
(2↑7) = 128

Theorem127prm 41840 127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.)
127 ∈ ℙ

Theoremm7prm 41841 The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.)
((2↑7) − 1) ∈ ℙ

Theorem2exp11 41842 Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
(2↑11) = 2048

Theoremm11nprm 41843 The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.)
((2↑11) − 1) = (89 · 23)

Theoremmod42tp1mod8 41844 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)

Theoremsfprmdvdsmersenne 41845 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))

Theoremsgprmdvdsmersenne 41846 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))

Theoremlighneallem1 41847 Lemma 1 for lighneal 41853. (Contributed by AV, 11-Aug-2021.)
((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃𝑀))

Theoremlighneallem2 41848 Lemma 2 for lighneal 41853. (Contributed by AV, 13-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneallem3 41849 Lemma 3 for lighneal 41853. (Contributed by AV, 11-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneallem4a 41850 Lemma 1 for lighneallem4 41852. (Contributed by AV, 16-Aug-2021.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘3) ∧ 𝑆 = (((𝐴𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆)

Theoremlighneallem4b 41851* Lemma 2 for lighneallem4 41852. (Contributed by AV, 16-Aug-2021.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴𝑘)) ∈ (ℤ‘2))

Theoremlighneallem4 41852 Lemma 3 for lighneal 41853. (Contributed by AV, 16-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneal 41853 If a power of a prime 𝑃 (i.e. 𝑃𝑀) is of the form 2↑𝑁 − 1, then 𝑁 must be prime and 𝑀 must be 1. Generalization of mersenne 24997 (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) = (𝑃𝑀)) → (𝑀 = 1 ∧ 𝑁 ∈ ℙ))

20.35.7.3  Proth's theorem

Theoremmodexp2m1d 41854 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.)
(𝜑𝐴 ∈ ℤ)    &   (𝜑𝐸 ∈ ℝ+)    &   (𝜑 → 1 < 𝐸)    &   (𝜑 → (𝐴 mod 𝐸) = (-1 mod 𝐸))       (𝜑 → ((𝐴↑2) mod 𝐸) = 1)

Theoremproththdlem 41855 Lemma for proththd 41856. (Contributed by AV, 4-Jul-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑃 = ((𝐾 · (2↑𝑁)) + 1))       (𝜑 → (𝑃 ∈ ℕ ∧ 1 < 𝑃 ∧ ((𝑃 − 1) / 2) ∈ ℕ))

Theoremproththd 41856* 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 15657), Proth's theorem allows for a convenient method for verifying large primes. (Contributed by AV, 5-Jul-2020.)
(𝜑𝑁 ∈ ℕ)    &   (𝜑𝐾 ∈ ℕ)    &   (𝜑𝑃 = ((𝐾 · (2↑𝑁)) + 1))    &   (𝜑𝐾 < (2↑𝑁))    &   (𝜑 → ∃𝑥 ∈ ℤ ((𝑥↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃))       (𝜑𝑃 ∈ ℙ)

Theorem5tcu2e40 41857 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
(5 · (2↑3)) = 40

Theorem3exp4mod41 41858 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
((3↑4) mod 41) = (-1 mod 41)

Theorem41prothprmlem1 41859 Lemma 1 for 41prothprm 41861. (Contributed by AV, 4-Jul-2020.)
𝑃 = 41       ((𝑃 − 1) / 2) = 20

Theorem41prothprmlem2 41860 Lemma 2 for 41prothprm 41861. (Contributed by AV, 5-Jul-2020.)
𝑃 = 41       ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)

Theorem41prothprm 41861 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
𝑃 = 41       (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ)

20.35.8  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 41864 and df-odd 41865. Alternate definitions resp. charaterizations are provided in dfeven2 41887, dfeven3 41895, dfeven4 41876 and in dfodd2 41874, dfodd3 41888, dfodd4 41896, dfodd5 41897, dfodd6 41875. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 41875 in opoeALTV 41919 and dfodd3 41888 in oddprmALTV 41923. 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 41918 and divgcdodd 15469).

20.35.8.1  Definitions and basic properties

Syntaxceven 41862 Extend the definition of a class to include the set of even numbers.
class Even

Syntaxcodd 41863 Extend the definition of a class to include the set of odd numbers.
class Odd

Definitiondf-even 41864 Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}

Definitiondf-odd 41865 Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}

Theoremiseven 41866 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))

Theoremisodd 41867 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))

Theoremevenz 41868 An even number is an integer. (Contributed by AV, 14-Jun-2020.)
(𝑍 ∈ Even → 𝑍 ∈ ℤ)

Theoremoddz 41869 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
(𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Theoremevendiv2z 41870 The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
(𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Theoremoddp1div2z 41871 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)

Theoremoddm1div2z 41872 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)

Theoremisodd2 41873 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))

Theoremdfodd2 41874 Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}

Theoremdfodd6 41875* Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}

Theoremdfeven4 41876* Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}

Theoremevenm1odd 41877 The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )

Theoremevenp1odd 41878 The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )

Theoremoddp1eveni 41879 The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )

Theoremoddm1eveni 41880 The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.)
(𝑍 ∈ Odd → (𝑍 − 1) ∈ Even )

Theoremevennodd 41881 An even number is not an odd number. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → ¬ 𝑍 ∈ Odd )

Theoremoddneven 41882 An odd number is not an even number. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Odd → ¬ 𝑍 ∈ Even )

Theoremenege 41883 The negative of an even number is even. (Contributed by AV, 20-Jun-2020.)
(𝐴 ∈ Even → -𝐴 ∈ Even )

Theoremonego 41884 The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.)
(𝐴 ∈ Odd → -𝐴 ∈ Odd )

Theoremm1expevenALTV 41885 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)

Theoremm1expoddALTV 41886 Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.)
(𝑁 ∈ Odd → (-1↑𝑁) = -1)

20.35.8.2  Alternate definitions using the "divides" relation

Theoremdfeven2 41887 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧}

Theoremdfodd3 41888 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧}

Theoremiseven2 41889 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))

Theoremisodd3 41890 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))

Theorem2dvdseven 41891 2 divides an even number. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Even → 2 ∥ 𝑍)

Theorem2ndvdsodd 41892 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → ¬ 2 ∥ 𝑍)

Theorem2dvdsoddp1 41893 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → 2 ∥ (𝑍 + 1))

Theorem2dvdsoddm1 41894 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.)
(𝑍 ∈ Odd → 2 ∥ (𝑍 − 1))

20.35.8.3  Alternate definitions using the "modulo" operation

Theoremdfeven3 41895 Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0}

Theoremdfodd4 41896 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1}

Theoremdfodd5 41897 Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0}

Theoremzefldiv2ALTV 41898 The floor of an even number divided by 2 is equal to the even number divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
(𝑁 ∈ Even → (⌊‘(𝑁 / 2)) = (𝑁 / 2))

Theoremzofldiv2ALTV 41899 The floor of an odd numer divided by 2 is equal to the odd number first decreased by 1 and then divided by 2. (Contributed by AV, 7-Jun-2020.) (Revised by AV, 18-Jun-2020.)
(𝑁 ∈ Odd → (⌊‘(𝑁 / 2)) = ((𝑁 − 1) / 2))

TheoremoddflALTV 41900 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 18-Jun-2020.)
(𝐾 ∈ Odd → 𝐾 = ((2 · (⌊‘(𝐾 / 2))) + 1))

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