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Theorem List for Metamath Proof Explorer - 39901-40000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfmtnorec2 39901* The second recurrence relation for Fermat numbers, see ProofWiki "Product of Sequence of Fermat Numbers plus 2", 29-Jul-2021, https://proofwiki.org/wiki/Product_of_Sequence_of_Fermat_Numbers_plus_2 or Wikipedia "Fermat number", 29-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = (∏𝑛 ∈ (0...𝑁)(FermatNo‘𝑛) + 2))

Theoremfmtnodvds 39902 Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 39901, see ProofWiki "Product of Sequence of Fermat Numbers plus 2/Corollary", 31-Jul-2021. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ) → (FermatNo‘𝑁) ∥ ((FermatNo‘(𝑁 + 𝑀)) − 2))

Theoremgoldbachthlem1 39903 Lemma 1 for goldbachth 39905. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))

Theoremgoldbachthlem2 39904 Lemma 2 for goldbachth 39905. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremgoldbachth 39905 Goldbach's theorem: Two different Fermat numbers are coprime. See ProofWiki "Goldbach's theorem", 31-Jul-2021, https://proofwiki.org/wiki/Goldbach%27s_Theorem or Wikipedia "Fermat number", 31-Jul-2021, https://en.wikipedia.org/wiki/Fermat_number#Basic_properties. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑁𝑀) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremfmtnorec3 39906* 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))))

Theoremfmtnorec4 39907 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))))

Theoremfmtno2 39908 The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) = 17

Theoremfmtno3 39909 The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘3) = 257

Theoremfmtno4 39910 The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘4) = 65537

Theoremfmtno5lem1 39911 Lemma 1 for fmtno5 39915. (Contributed by AV, 22-Jul-2021.)
(65536 · 6) = 393216

Theoremfmtno5lem2 39912 Lemma 2 for fmtno5 39915. (Contributed by AV, 22-Jul-2021.)
(65536 · 5) = 327680

Theoremfmtno5lem3 39913 Lemma 3 for fmtno5 39915. (Contributed by AV, 22-Jul-2021.)
(65536 · 3) = 196608

Theoremfmtno5lem4 39914 Lemma 4 for fmtno5 39915. (Contributed by AV, 30-Jul-2021.)
(65536↑2) = 4294967296

Theoremfmtno5 39915 The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
(FermatNo‘5) = 4294967297

Theoremfmtno0prm 39916 The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) ∈ ℙ

Theoremfmtno1prm 39917 The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) ∈ ℙ

Theoremfmtno2prm 39918 The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) ∈ ℙ

Theorem257prm 39919 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
257 ∈ ℙ

Theoremfmtno3prm 39920 The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
(FermatNo‘3) ∈ ℙ

Theoremodz2prm2pw 39921 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)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))

Theoremfmtnoprmfac1lem 39922 Lemma for fmtnoprmfac1 39923: The order of 2 modulo a prime that divides the n-th Fermat number is 2^(n+1). (Contributed by AV, 25-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((od𝑃)‘2) = (2↑(𝑁 + 1)))

Theoremfmtnoprmfac1 39923* Divisor of Fermat number (special form of Euler's result, see fmtnofac1 39928): 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 39924 Lemma for fmtnoprmfac2 39925. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)

Theoremfmtnoprmfac2 39925* Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 39927): 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 39926* Lemma for fmtnofac2 39927 (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 39927* Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 39928: 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 39928* 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 39927. (Contributed by AV, 30-Jul-2021.)

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

Theoremfmtno4sqrt 39929 The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
(⌊‘(√‘(FermatNo‘4))) = 256

Theoremfmtno4prmfac 39930 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 39931 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 39932 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
¬ 193 ∥ (FermatNo‘4)

Theoremfmtno4prm 39933 The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
(FermatNo‘4) ∈ ℙ

Theorem65537prm 39934 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
65537 ∈ ℙ

Theoremfmtnofz04prm 39935 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 39936 The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ0𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)

Theoremfmtno5faclem1 39937 Lemma 1 for fmtno5fac 39940. (Contributed by AV, 22-Jul-2021.)
(6700417 · 4) = 26801668

Theoremfmtno5faclem2 39938 Lemma 2 for fmtno5fac 39940. (Contributed by AV, 22-Jul-2021.)
(6700417 · 6) = 40202502

Theoremfmtno5faclem3 39939 Lemma 3 for fmtno5fac 39940. (Contributed by AV, 22-Jul-2021.)
(402025020 + 26801668) = 428826688

Theoremfmtno5fac 39940 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 39941 The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
(FermatNo‘5) ∉ ℙ

Theoremprmdvdsfmtnof1lem1 39942* Lemma 1 for prmdvdsfmtnof1 39945. (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 39943 Lemma 2 for prmdvdsfmtnof1 39945. (Contributed by AV, 3-Aug-2021.)
((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))

Theoremprmdvdsfmtnof 39944* 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 39945* 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 39946 The set of prime numbers is infinite. The proof of this variant of prminf 15341 is based on Goldbach's theorem goldbachth 39905 (via prmdvdsfmtnof1 39945 and prmdvdsfmtnof1lem2 39943), 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 39947* The difference of two numbers to the same power is the difference of the two numbers multiplied with a finite sum. Generalization of subsq 12702. 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 39948* 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 14308 is not based on geoser 14307, but on pwdif 39947 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 39949* 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 39950* 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))

20.34.4.2  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 24641. 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 39964. This is an example of sgprmdvdsmersenne 39967, 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 39966, 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 39974.

Theoremm2prm 39951 The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑2) − 1) ∈ ℙ

Theoremm3prm 39952 The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑3) − 1) ∈ ℙ

Theorem2exp5 39953 Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
(2↑5) = 32

Theoremflsqrt 39954 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 39955 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 39956 3 does not divide 4. (Contributed by AV, 18-Aug-2021.)
¬ 3 ∥ 4

Theorem139prmALT 39957 139 is a prime number. In contrast to 139prm 15553, the proof of this theorem uses 3dvds2dec 14763 for checking the divisibility by 3. Although the proof using 3dvds2dec 14763 is longer (regarding size: 1849 characters compared with 1809 for 139prm 15553), the number of essential steps is smaller (301 compared with 327 for 139prm 15553). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
139 ∈ ℙ

Theorem31prm 39958 31 is a prime number. In contrast to 37prm 15550, the proof of this theorem is not based on the "blanket" prmlem2 15549, but on isprm7 15134. 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 15550 (1810 characters compared with 1213 for 37prm 15550). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 15550). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.)
31 ∈ ℙ

Theoremm5prm 39959 The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.)
((2↑5) − 1) ∈ ℙ

Theorem2exp7 39960 Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
(2↑7) = 128

Theorem127prm 39961 127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.)
127 ∈ ℙ

Theoremm7prm 39962 The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.)
((2↑7) − 1) ∈ ℙ

Theorem2exp11 39963 Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
(2↑11) = 2048

Theoremm11nprm 39964 The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.)
((2↑11) − 1) = (89 · 23)

Theoremmod42tp1mod8 39965 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 39966 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 39967 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 39968 Lemma 1 for lighneal 39974. (Contributed by AV, 11-Aug-2021.)
((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃𝑀))

Theoremlighneallem2 39969 Lemma 2 for lighneal 39974. (Contributed by AV, 13-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneallem3 39970 Lemma 3 for lighneal 39974. (Contributed by AV, 11-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneallem4a 39971 Lemma 1 for lighneallem4 39973. (Contributed by AV, 16-Aug-2021.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘3) ∧ 𝑆 = (((𝐴𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆)

Theoremlighneallem4b 39972* Lemma 2 for lighneallem4 39973. (Contributed by AV, 16-Aug-2021.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴𝑘)) ∈ (ℤ‘2))

Theoremlighneallem4 39973 Lemma 3 for lighneal 39974. (Contributed by AV, 16-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneal 39974 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 24641 (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))

20.34.4.3  Proth's theorem

Theoremmodexp2m1d 39975 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 39976 Lemma for proththd 39977. (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 39977* 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 15332), 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 39978 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
(5 · (2↑3)) = 40

Theorem3exp4mod41 39979 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
((3↑4) mod 41) = (-1 mod 41)

Theorem41prothprmlem1 39980 Lemma 1 for 41prothprm 39982. (Contributed by AV, 4-Jul-2020.)
𝑃 = 41       ((𝑃 − 1) / 2) = 20

Theorem41prothprmlem2 39981 Lemma 2 for 41prothprm 39982. (Contributed by AV, 5-Jul-2020.)
𝑃 = 41       ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)

Theorem41prothprm 39982 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.)
𝑃 = 41       (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ)

20.34.5  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 39985 and df-odd 39986. Alternate definitions resp. charaterizations are provided in dfeven2 40008, dfeven3 40016, dfeven4 39997 and in dfodd2 39995, dfodd3 40009, dfodd4 40017, dfodd5 40018, dfodd6 39996. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 39996 in opoeALTV 40040 and dfodd3 40009 in oddprmALTV 40044. 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 40039 and divgcdodd 15136).

20.34.5.1  Definitions and basic properties

Syntaxceven 39983 Extend the definition of a class to include the set of even numbers.
class Even

Syntaxcodd 39984 Extend the definition of a class to include the set of odd numbers.
class Odd

Definitiondf-even 39985 Define the set of even numbers. (Contributed by AV, 14-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ}

Definitiondf-odd 39986 Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ}

Theoremiseven 39987 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 39988 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 39989 An even number is an integer. (Contributed by AV, 14-Jun-2020.)
(𝑍 ∈ Even → 𝑍 ∈ ℤ)

Theoremoddz 39990 An odd number is an integer. (Contributed by AV, 14-Jun-2020.)
(𝑍 ∈ Odd → 𝑍 ∈ ℤ)

Theoremevendiv2z 39991 The result of dividing an even number by 2 is an integer. (Contributed by AV, 15-Jun-2020.)
(𝑍 ∈ Even → (𝑍 / 2) ∈ ℤ)

Theoremoddp1div2z 39992 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 39993 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 39994 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 39995 Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ}

Theoremdfodd6 39996* Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.)
Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)}

Theoremdfeven4 39997* Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.)
Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)}

Theoremevenm1odd 39998 The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → (𝑍 − 1) ∈ Odd )

Theoremevenp1odd 39999 The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Even → (𝑍 + 1) ∈ Odd )

Theoremoddp1eveni 40000 The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.)
(𝑍 ∈ Odd → (𝑍 + 1) ∈ Even )

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