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Theorem List for Metamath Proof Explorer - 41201-41300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremrepswpfx 41201 A prefix of a repeated symbol word is a repeated symbol word. (Contributed by AV, 11-May-2020.)
((𝑆𝑉𝑁 ∈ ℕ0𝐿 ∈ (0...𝑁)) → ((𝑆 repeatS 𝑁) prefix 𝐿) = (𝑆 repeatS 𝐿))

Theoremcshword2 41202 Perform a cyclical shift for a word. (Contributed by AV, 11-May-2020.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℤ) → (𝑊 cyclShift 𝑁) = ((𝑊 substr ⟨(𝑁 mod (#‘𝑊)), (#‘𝑊)⟩) ++ (𝑊 prefix (𝑁 mod (#‘𝑊)))))

Theorempfxco 41203 Mapping of words commutes with the prefix operation. (Contributed by AV, 15-May-2020.)
((𝑊 ∈ Word 𝐴𝑁 ∈ (0...(#‘𝑊)) ∧ 𝐹:𝐴𝐵) → (𝐹 ∘ (𝑊 prefix 𝑁)) = ((𝐹𝑊) prefix 𝑁))

20.34.7  Number theory (extension)

20.34.7.1  Fermat numbers

At first, the (sequence of) Fermat numbers FermatNo (the 𝑛-th Fermat number is denoted as (FermatNo‘𝑛)) is defined, see df-fmtno 41205, and basic theorems are provided. Afterwards, it is shown that the first five Fermat numbers are prime, the (first) five Fermat primes, see fmtnofz04prm 41254, but that the fifth Fermat number (counting starts at 0!) is not prime, see fmtno5nprm 41260. The fourth Fermat number (i.e., the fifth Fermat prime) (FermatNo‘4) = 65537 is currently the biggest number proven to be prime in set.mm, see 65537prm 41253 (previously, it was 4001, see 4001prm 15833).

Another important result of this section is Goldbach's theorem goldbachth 41224, showing that two different Fermut numbers are coprime. By this, it can be proven that there is an infinite number of primes, see prminf2 41265.

Finally, it is shown that every prime of the form ((2↑𝑘) + 1) must be a Fermat number (i.e., a Fermat prime), see 2pwp1prmfmtno 41269.

Syntaxcfmtno 41204 Extend class notation with the Fermat numbers.
class FermatNo

Definitiondf-fmtno 41205 Define the function that enumerates the Fermat numbers, see definition in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
FermatNo = (𝑛 ∈ ℕ0 ↦ ((2↑(2↑𝑛)) + 1))

Theoremfmtno 41206 The 𝑁 th Fermat number. (Contributed by AV, 13-Jun-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) = ((2↑(2↑𝑁)) + 1))

Theoremfmtnoge3 41207 Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ (ℤ‘3))

Theoremfmtnonn 41208 Each Fermat number is a positive integer. (Contributed by AV, 26-Jul-2021.) (Proof shortened by AV, 4-Aug-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘𝑁) ∈ ℕ)

Theoremfmtnom1nn 41209 A Fermat number minus one is a power of a power of two. (Contributed by AV, 29-Jul-2021.)
(𝑁 ∈ ℕ0 → ((FermatNo‘𝑁) − 1) = (2↑(2↑𝑁)))

Theoremfmtnoodd 41210 Each Fermat number is odd. (Contributed by AV, 26-Jul-2021.)
(𝑁 ∈ ℕ0 → ¬ 2 ∥ (FermatNo‘𝑁))

Theoremfmtnorn 41211* A Fermat number is a function value of the enumeration of the Fermat numbers. (Contributed by AV, 3-Aug-2021.)
(𝐹 ∈ ran FermatNo ↔ ∃𝑛 ∈ ℕ0 (FermatNo‘𝑛) = 𝐹)

Theoremfmtnof1 41212 The enumeration of the Fermat numbers is a one-one function into the positive integers. (Contributed by AV, 3-Aug-2021.)
FermatNo:ℕ01-1→ℕ

Theoremfmtnoinf 41213 The set of Fermat numbers is infinite. (Contributed by AV, 3-Aug-2021.)
ran FermatNo ∉ Fin

Theoremfmtnorec1 41214 The first recurrence relation for Fermat numbers, see Wikipedia "Fermat number", https://en.wikipedia.org/wiki/Fermat_number#Basic_properties, 22-Jul-2021. (Contributed by AV, 22-Jul-2021.)
(𝑁 ∈ ℕ0 → (FermatNo‘(𝑁 + 1)) = ((((FermatNo‘𝑁) − 1)↑2) + 1))

Theoremsqrtpwpw2p 41215 The floor of the square root of 2 to the power of 2 to the power of a positive integer plus a bounded nonnegative integer. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ ∧ 𝑀 ∈ ℕ0𝑀 < ((2↑((2↑(𝑁 − 1)) + 1)) + 1)) → (⌊‘(√‘((2↑(2↑𝑁)) + 𝑀))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtnosqrt 41216 The floor of the square root of a Fermat number. (Contributed by AV, 28-Jul-2021.)
(𝑁 ∈ ℕ → (⌊‘(√‘(FermatNo‘𝑁))) = (2↑(2↑(𝑁 − 1))))

Theoremfmtno0 41217 The 0 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) = 3

Theoremfmtno1 41218 The 1 st Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) = 5

Theoremfmtnorec2lem 41219* Lemma for fmtnorec2 41220 (induction step). (Contributed by AV, 29-Jul-2021.)
(𝑦 ∈ ℕ0 → ((FermatNo‘(𝑦 + 1)) = (∏𝑛 ∈ (0...𝑦)(FermatNo‘𝑛) + 2) → (FermatNo‘((𝑦 + 1) + 1)) = (∏𝑛 ∈ (0...(𝑦 + 1))(FermatNo‘𝑛) + 2)))

Theoremfmtnorec2 41220* 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 41221 Any Fermat number divides a greater Fermat number minus 2. Corrolary of fmtnorec2 41220, 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 41222 Lemma 1 for goldbachth 41224. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → (FermatNo‘𝑀) ∥ ((FermatNo‘𝑁) − 2))

Theoremgoldbachthlem2 41223 Lemma 2 for goldbachth 41224. (Contributed by AV, 1-Aug-2021.)
((𝑁 ∈ ℕ0𝑀 ∈ ℕ0𝑀 < 𝑁) → ((FermatNo‘𝑁) gcd (FermatNo‘𝑀)) = 1)

Theoremgoldbachth 41224 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 41225* 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 41226 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 41227 The 2 nd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) = 17

Theoremfmtno3 41228 The 3 rd Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘3) = 257

Theoremfmtno4 41229 The 4 th Fermat number, see remark in [ApostolNT] p. 7. (Contributed by AV, 13-Jun-2021.)
(FermatNo‘4) = 65537

Theoremfmtno5lem1 41230 Lemma 1 for fmtno5 41234. (Contributed by AV, 22-Jul-2021.)
(65536 · 6) = 393216

Theoremfmtno5lem2 41231 Lemma 2 for fmtno5 41234. (Contributed by AV, 22-Jul-2021.)
(65536 · 5) = 327680

Theoremfmtno5lem3 41232 Lemma 3 for fmtno5 41234. (Contributed by AV, 22-Jul-2021.)
(65536 · 3) = 196608

Theoremfmtno5lem4 41233 Lemma 4 for fmtno5 41234. (Contributed by AV, 30-Jul-2021.)
(65536↑2) = 4294967296

Theoremfmtno5 41234 The 5 th Fermat number. (Contributed by AV, 30-Jul-2021.)
(FermatNo‘5) = 4294967297

Theoremfmtno0prm 41235 The 0 th Fermat number is a prime (first Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘0) ∈ ℙ

Theoremfmtno1prm 41236 The 1 st Fermat number is a prime (second Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘1) ∈ ℙ

Theoremfmtno2prm 41237 The 2 nd Fermat number is a prime (third Fermat prime). (Contributed by AV, 13-Jun-2021.)
(FermatNo‘2) ∈ ℙ

Theorem257prm 41238 257 is a prime number (the fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
257 ∈ ℙ

Theoremfmtno3prm 41239 The 3 rd Fermat number is a prime (fourth Fermat prime). (Contributed by AV, 15-Jun-2021.)
(FermatNo‘3) ∈ ℙ

Theoremodz2prm2pw 41240 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 41241 Lemma for fmtnoprmfac1 41242: 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 41242* Divisor of Fermat number (special form of Euler's result, see fmtnofac1 41247): 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 41243 Lemma for fmtnoprmfac2 41244. (Contributed by AV, 26-Jul-2021.)
((𝑁 ∈ (ℤ‘2) ∧ 𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑃 ∥ (FermatNo‘𝑁)) → ((2↑((𝑃 − 1) / 2)) mod 𝑃) = 1)

Theoremfmtnoprmfac2 41244* Divisor of Fermat number (special form of Lucas' result, see fmtnofac2 41246): 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 41245* Lemma for fmtnofac2 41246 (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 41246* Divisor of Fermat number (Euler's Result refined by François Édouard Anatole Lucas), see fmtnofac1 41247: 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 41247* 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 41246. (Contributed by AV, 30-Jul-2021.)

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

Theoremfmtno4sqrt 41248 The floor of the square root of the fourth Fermat number is 256. (Contributed by AV, 28-Jul-2021.)
(⌊‘(√‘(FermatNo‘4))) = 256

Theoremfmtno4prmfac 41249 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 41250 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 41251 193 is not a (prime) factor of the fourth Fermat number. (Contributed by AV, 24-Jul-2021.)
¬ 193 ∥ (FermatNo‘4)

Theoremfmtno4prm 41252 The 4-th Fermat number (65537) is a prime (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
(FermatNo‘4) ∈ ℙ

Theorem65537prm 41253 65537 is a prime number (the fifth Fermat prime). (Contributed by AV, 28-Jul-2021.)
65537 ∈ ℙ

Theoremfmtnofz04prm 41254 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 41255 The first five Fermat numbers are prime. (Contributed by AV, 28-Jul-2021.)
((𝑁 ∈ ℕ0𝑁 ≤ 4) → (FermatNo‘𝑁) ∈ ℙ)

Theoremfmtno5faclem1 41256 Lemma 1 for fmtno5fac 41259. (Contributed by AV, 22-Jul-2021.)
(6700417 · 4) = 26801668

Theoremfmtno5faclem2 41257 Lemma 2 for fmtno5fac 41259. (Contributed by AV, 22-Jul-2021.)
(6700417 · 6) = 40202502

Theoremfmtno5faclem3 41258 Lemma 3 for fmtno5fac 41259. (Contributed by AV, 22-Jul-2021.)
(402025020 + 26801668) = 428826688

Theoremfmtno5fac 41259 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 41260 The 5 th Fermat number is a not a prime. (Contributed by AV, 22-Jul-2021.)
(FermatNo‘5) ∉ ℙ

Theoremprmdvdsfmtnof1lem1 41261* Lemma 1 for prmdvdsfmtnof1 41264. (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 41262 Lemma 2 for prmdvdsfmtnof1 41264. (Contributed by AV, 3-Aug-2021.)
((𝐹 ∈ ran FermatNo ∧ 𝐺 ∈ ran FermatNo) → ((𝐼 ∈ ℙ ∧ 𝐼𝐹𝐼𝐺) → 𝐹 = 𝐺))

Theoremprmdvdsfmtnof 41263* 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 41264* 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 41265 The set of prime numbers is infinite. The proof of this variant of prminf 15600 is based on Goldbach's theorem goldbachth 41224 (via prmdvdsfmtnof1 41264 and prmdvdsfmtnof1lem2 41262), 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 41266* The difference of two numbers to the same power is the difference of the two numbers multiplied with a finite sum. Generalization of subsq 12955. 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 41267* 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 14581 is not based on geoser 14580, but on pwdif 41266 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 41268* 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 41269* 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))

20.34.7.2  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 24933. 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 41283. This is an example of sgprmdvdsmersenne 41286, 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 41285, 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 41293.

Theoremm2prm 41270 The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑2) − 1) ∈ ℙ

Theoremm3prm 41271 The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.)
((2↑3) − 1) ∈ ℙ

Theorem2exp5 41272 Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.)
(2↑5) = 32

Theoremflsqrt 41273 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 41274 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 41275 3 does not divide 4. (Contributed by AV, 18-Aug-2021.)
¬ 3 ∥ 4

Theorem139prmALT 41276 139 is a prime number. In contrast to 139prm 15812, the proof of this theorem uses 3dvds2dec 15037 for checking the divisibility by 3. Although the proof using 3dvds2dec 15037 is longer (regarding size: 1849 characters compared with 1809 for 139prm 15812), the number of essential steps is smaller (301 compared with 327 for 139prm 15812). (Contributed by Mario Carneiro, 19-Feb-2014.) (Revised by AV, 18-Aug-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
139 ∈ ℙ

Theorem31prm 41277 31 is a prime number. In contrast to 37prm 15809, the proof of this theorem is not based on the "blanket" prmlem2 15808, but on isprm7 15401. 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 15809 (1810 characters compared with 1213 for 37prm 15809). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 15809). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.)
31 ∈ ℙ

Theoremm5prm 41278 The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.)
((2↑5) − 1) ∈ ℙ

Theorem2exp7 41279 Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.)
(2↑7) = 128

Theorem127prm 41280 127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.)
127 ∈ ℙ

Theoremm7prm 41281 The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.)
((2↑7) − 1) ∈ ℙ

Theorem2exp11 41282 Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.)
(2↑11) = 2048

Theoremm11nprm 41283 The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.)
((2↑11) − 1) = (89 · 23)

Theoremmod42tp1mod8 41284 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 41285 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 41286 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 41287 Lemma 1 for lighneal 41293. (Contributed by AV, 11-Aug-2021.)
((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃𝑀))

Theoremlighneallem2 41288 Lemma 2 for lighneal 41293. (Contributed by AV, 13-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneallem3 41289 Lemma 3 for lighneal 41293. (Contributed by AV, 11-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneallem4a 41290 Lemma 1 for lighneallem4 41292. (Contributed by AV, 16-Aug-2021.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘3) ∧ 𝑆 = (((𝐴𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆)

Theoremlighneallem4b 41291* Lemma 2 for lighneallem4 41292. (Contributed by AV, 16-Aug-2021.)
((𝐴 ∈ (ℤ‘2) ∧ 𝑀 ∈ (ℤ‘2) ∧ ¬ 2 ∥ 𝑀) → Σ𝑘 ∈ (0...(𝑀 − 1))((-1↑𝑘) · (𝐴𝑘)) ∈ (ℤ‘2))

Theoremlighneallem4 41292 Lemma 3 for lighneal 41293. (Contributed by AV, 16-Aug-2021.)
(((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (¬ 2 ∥ 𝑁 ∧ ¬ 2 ∥ 𝑀) ∧ ((2↑𝑁) − 1) = (𝑃𝑀)) → 𝑀 = 1)

Theoremlighneal 41293 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 24933 (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))

20.34.7.3  Proth's theorem

Theoremmodexp2m1d 41294 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 41295 Lemma for proththd 41296. (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 41296* 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 15591), 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 41297 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.)
(5 · (2↑3)) = 40

Theorem3exp4mod41 41298 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.)
((3↑4) mod 41) = (-1 mod 41)

Theorem41prothprmlem1 41299 Lemma 1 for 41prothprm 41301. (Contributed by AV, 4-Jul-2020.)
𝑃 = 41       ((𝑃 − 1) / 2) = 20

Theorem41prothprmlem2 41300 Lemma 2 for 41prothprm 41301. (Contributed by AV, 5-Jul-2020.)
𝑃 = 41       ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃)

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