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Type | Label | Description |
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Statement | ||
Theorem | pwdif 41501* | The difference of two numbers to the same power is the difference of the two numbers multiplied with a finite sum. Generalization of subsq 12972. 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 ∈ ℕ0 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑𝑁) − (𝐵↑𝑁)) = ((𝐴 − 𝐵) · Σ𝑘 ∈ (0..^𝑁)((𝐴↑𝑘) · (𝐵↑((𝑁 − 𝑘) − 1))))) | ||
Theorem | pwm1geoserALT 41502* | 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 14600 is not based on geoser 14599, but on pwdif 41501 and therefore shorter than the original proof. (Contributed by AV, 19-Aug-2021.) (Proof modification is discouraged.) (New usage is discouraged.) |
⊢ (𝜑 → 𝐴 ∈ ℂ) & ⊢ (𝜑 → 𝑁 ∈ ℕ0) ⇒ ⊢ (𝜑 → ((𝐴↑𝑁) − 1) = ((𝐴 − 1) · Σ𝑘 ∈ (0...(𝑁 − 1))(𝐴↑𝑘))) | ||
Theorem | 2pwp1prm 41503* | 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)) | ||
Theorem | 2pwp1prmfmtno 41504* | 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)) | ||
"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 24952. 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 41518. This is an example of sgprmdvdsmersenne 41521, 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 41520, 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 41528. | ||
Theorem | m2prm 41505 | The second Mersenne number M2 = 3 is a prime number. (Contributed by AV, 16-Aug-2021.) |
⊢ ((2↑2) − 1) ∈ ℙ | ||
Theorem | m3prm 41506 | The third Mersenne number M3 = 7 is a prime number. (Contributed by AV, 16-Aug-2021.) |
⊢ ((2↑3) − 1) ∈ ℙ | ||
Theorem | 2exp5 41507 | Two to the fifth power is 32. (Contributed by AV, 16-Aug-2021.) |
⊢ (2↑5) = ;32 | ||
Theorem | flsqrt 41508 | 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 41509 | 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 41510 | 3 does not divide 4. (Contributed by AV, 18-Aug-2021.) |
⊢ ¬ 3 ∥ 4 | ||
Theorem | 139prmALT 41511 | 139 is a prime number. In contrast to 139prm 15831, the proof of this theorem uses 3dvds2dec 15056 for checking the divisibility by 3. Although the proof using 3dvds2dec 15056 is longer (regarding size: 1849 characters compared with 1809 for 139prm 15831), the number of essential steps is smaller (301 compared with 327 for 139prm 15831). (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 41512 | 31 is a prime number. In contrast to 37prm 15828, the proof of this theorem is not based on the "blanket" prmlem2 15827, but on isprm7 15420. 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 15828 (1810 characters compared with 1213 for 37prm 15828). The number of essential steps, however, is much smaller (138 compared with 213 for 37prm 15828). (Contributed by AV, 17-Aug-2021.) (Proof modification is discouraged.) |
⊢ ;31 ∈ ℙ | ||
Theorem | m5prm 41513 | The fifth Mersenne number M5 = 31 is a prime number. (Contributed by AV, 17-Aug-2021.) |
⊢ ((2↑5) − 1) ∈ ℙ | ||
Theorem | 2exp7 41514 | Two to the seventh power is 128. (Contributed by AV, 16-Aug-2021.) |
⊢ (2↑7) = ;;128 | ||
Theorem | 127prm 41515 | 127 is a prime number. (Contributed by AV, 16-Aug-2021.) (Proof shortened by AV, 16-Sep-2021.) |
⊢ ;;127 ∈ ℙ | ||
Theorem | m7prm 41516 | The seventh Mersenne number M7 = 127 is a prime number. (Contributed by AV, 18-Aug-2021.) |
⊢ ((2↑7) − 1) ∈ ℙ | ||
Theorem | 2exp11 41517 | Two to the eleventh power is 2048. (Contributed by AV, 16-Aug-2021.) |
⊢ (2↑;11) = ;;;2048 | ||
Theorem | m11nprm 41518 | The eleventh Mersenne number M11 = 2047 is not a prime number. (Contributed by AV, 18-Aug-2021.) |
⊢ ((2↑;11) − 1) = (;89 · ;23) | ||
Theorem | mod42tp1mod8 41519 | 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 41520 | 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 41521 | 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 41522 | Lemma 1 for lighneal 41528. (Contributed by AV, 11-Aug-2021.) |
⊢ ((𝑃 = 2 ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((2↑𝑁) − 1) ≠ (𝑃↑𝑀)) | ||
Theorem | lighneallem2 41523 | Lemma 2 for lighneal 41528. (Contributed by AV, 13-Aug-2021.) |
⊢ (((𝑃 ∈ (ℙ ∖ {2}) ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ 2 ∥ 𝑁 ∧ ((2↑𝑁) − 1) = (𝑃↑𝑀)) → 𝑀 = 1) | ||
Theorem | lighneallem3 41524 | Lemma 3 for lighneal 41528. (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 41525 | Lemma 1 for lighneallem4 41527. (Contributed by AV, 16-Aug-2021.) |
⊢ ((𝐴 ∈ (ℤ≥‘2) ∧ 𝑀 ∈ (ℤ≥‘3) ∧ 𝑆 = (((𝐴↑𝑀) + 1) / (𝐴 + 1))) → 2 ≤ 𝑆) | ||
Theorem | lighneallem4b 41526* | Lemma 2 for lighneallem4 41527. (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 41527 | Lemma 3 for lighneal 41528. (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 41528 | 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 24952 (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)) | ||
Theorem | modexp2m1d 41529 | 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 41530 | Lemma for proththd 41531. (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 41531* | 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 15610), 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 41532 | 5 times the cube of 2 is 40. (Contributed by AV, 4-Jul-2020.) |
⊢ (5 · (2↑3)) = ;40 | ||
Theorem | 3exp4mod41 41533 | 3 to the fourth power is -1 modulo 41. (Contributed by AV, 5-Jul-2020.) |
⊢ ((3↑4) mod ;41) = (-1 mod ;41) | ||
Theorem | 41prothprmlem1 41534 | Lemma 1 for 41prothprm 41536. (Contributed by AV, 4-Jul-2020.) |
⊢ 𝑃 = ;41 ⇒ ⊢ ((𝑃 − 1) / 2) = ;20 | ||
Theorem | 41prothprmlem2 41535 | Lemma 2 for 41prothprm 41536. (Contributed by AV, 5-Jul-2020.) |
⊢ 𝑃 = ;41 ⇒ ⊢ ((3↑((𝑃 − 1) / 2)) mod 𝑃) = (-1 mod 𝑃) | ||
Theorem | 41prothprm 41536 | 41 is a Proth prime. (Contributed by AV, 5-Jul-2020.) |
⊢ 𝑃 = ;41 ⇒ ⊢ (𝑃 = ((5 · (2↑3)) + 1) ∧ 𝑃 ∈ ℙ) | ||
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 41539 and df-odd 41540. Alternate definitions resp. charaterizations are provided in dfeven2 41562, dfeven3 41570, dfeven4 41551 and in dfodd2 41549, dfodd3 41563, dfodd4 41571, dfodd5 41572, dfodd6 41550. Each characterization can be useful (and used) in an appropriate context, e.g. dfodd6 41550 in opoeALTV 41594 and dfodd3 41563 in oddprmALTV 41598. 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 41593 and divgcdodd 15422). | ||
Syntax | ceven 41537 | Extend the definition of a class to include the set of even numbers. |
class Even | ||
Syntax | codd 41538 | Extend the definition of a class to include the set of odd numbers. |
class Odd | ||
Definition | df-even 41539 | Define the set of even numbers. (Contributed by AV, 14-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 / 2) ∈ ℤ} | ||
Definition | df-odd 41540 | Define the set of odd numbers. (Contributed by AV, 14-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 + 1) / 2) ∈ ℤ} | ||
Theorem | iseven 41541 | 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 41542 | 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 41543 | An even number is an integer. (Contributed by AV, 14-Jun-2020.) |
⊢ (𝑍 ∈ Even → 𝑍 ∈ ℤ) | ||
Theorem | oddz 41544 | An odd number is an integer. (Contributed by AV, 14-Jun-2020.) |
⊢ (𝑍 ∈ Odd → 𝑍 ∈ ℤ) | ||
Theorem | evendiv2z 41545 | 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 41546 | 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 41547 | 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 41548 | 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 41549 | Alternate definition for odd numbers. (Contributed by AV, 15-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ((𝑧 − 1) / 2) ∈ ℤ} | ||
Theorem | dfodd6 41550* | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = ((2 · 𝑖) + 1)} | ||
Theorem | dfeven4 41551* | Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ ∃𝑖 ∈ ℤ 𝑧 = (2 · 𝑖)} | ||
Theorem | evenm1odd 41552 | The predecessor of an even number is odd. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Even → (𝑍 − 1) ∈ Odd ) | ||
Theorem | evenp1odd 41553 | The successor of an even number is odd. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Even → (𝑍 + 1) ∈ Odd ) | ||
Theorem | oddp1eveni 41554 | The successor of an odd number is even. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Odd → (𝑍 + 1) ∈ Even ) | ||
Theorem | oddm1eveni 41555 | The predecessor of an odd number is even. (Contributed by AV, 6-Jul-2020.) |
⊢ (𝑍 ∈ Odd → (𝑍 − 1) ∈ Even ) | ||
Theorem | evennodd 41556 | An even number is not an odd number. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Even → ¬ 𝑍 ∈ Odd ) | ||
Theorem | oddneven 41557 | An odd number is not an even number. (Contributed by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ Odd → ¬ 𝑍 ∈ Even ) | ||
Theorem | enege 41558 | The negative of an even number is even. (Contributed by AV, 20-Jun-2020.) |
⊢ (𝐴 ∈ Even → -𝐴 ∈ Even ) | ||
Theorem | onego 41559 | The negative of an odd number is odd. (Contributed by AV, 20-Jun-2020.) |
⊢ (𝐴 ∈ Odd → -𝐴 ∈ Odd ) | ||
Theorem | m1expevenALTV 41560 | 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 41561 | Exponentiation of -1 by an odd power. (Contributed by AV, 6-Jul-2020.) |
⊢ (𝑁 ∈ Odd → (-1↑𝑁) = -1) | ||
Theorem | dfeven2 41562 | Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ 2 ∥ 𝑧} | ||
Theorem | dfodd3 41563 | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ ¬ 2 ∥ 𝑧} | ||
Theorem | iseven2 41564 | 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 41565 | 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 41566 | 2 divides an even number. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Even → 2 ∥ 𝑍) | ||
Theorem | 2ndvdsodd 41567 | 2 does not divide an odd number. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Odd → ¬ 2 ∥ 𝑍) | ||
Theorem | 2dvdsoddp1 41568 | 2 divides an odd number increased by 1. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 + 1)) | ||
Theorem | 2dvdsoddm1 41569 | 2 divides an odd number decreased by 1. (Contributed by AV, 18-Jun-2020.) |
⊢ (𝑍 ∈ Odd → 2 ∥ (𝑍 − 1)) | ||
Theorem | dfeven3 41570 | Alternate definition for even numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 0} | ||
Theorem | dfodd4 41571 | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) = 1} | ||
Theorem | dfodd5 41572 | Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ (𝑧 mod 2) ≠ 0} | ||
Theorem | zefldiv2ALTV 41573 | 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)) | ||
Theorem | zofldiv2ALTV 41574 | 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)) | ||
Theorem | oddflALTV 41575 | 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)) | ||
Theorem | iseven5 41576 | The predicate "is an even number". An even number and 2 have 2 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) |
⊢ (𝑍 ∈ Even ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 2)) | ||
Theorem | isodd7 41577 | The predicate "is an odd number". An odd number and 2 have 1 as greatest common divisor. (Contributed by AV, 1-Jul-2020.) |
⊢ (𝑍 ∈ Odd ↔ (𝑍 ∈ ℤ ∧ (2 gcd 𝑍) = 1)) | ||
Theorem | dfeven5 41578 | Alternate definition for even numbers. (Contributed by AV, 1-Jul-2020.) |
⊢ Even = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 2} | ||
Theorem | dfodd7 41579 | Alternate definition for odd numbers. (Contributed by AV, 1-Jul-2020.) |
⊢ Odd = {𝑧 ∈ ℤ ∣ (2 gcd 𝑧) = 1} | ||
Theorem | zneoALTV 41580 | No even integer equals an odd integer (i.e. no integer can be both even and odd). Exercise 10(a) of [Apostol] p. 28. (Contributed by NM, 31-Jul-2004.) (Revised by AV, 16-Jun-2020.) |
⊢ ((𝐴 ∈ Even ∧ 𝐵 ∈ Odd ) → 𝐴 ≠ 𝐵) | ||
Theorem | zeoALTV 41581 | An integer is even or odd. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ Even ∨ 𝑍 ∈ Odd )) | ||
Theorem | zeo2ALTV 41582 | An integer is even or odd but not both. (Contributed by Mario Carneiro, 12-Sep-2015.) (Revised by AV, 16-Jun-2020.) |
⊢ (𝑍 ∈ ℤ → (𝑍 ∈ Even ↔ ¬ 𝑍 ∈ Odd )) | ||
Theorem | nneoALTV 41583 | A positive integer is even or odd but not both. (Contributed by NM, 1-Jan-2006.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ ℕ → (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd )) | ||
Theorem | nneoiALTV 41584 | A positive integer is even or odd but not both. (Contributed by NM, 20-Aug-2001.) (Revised by AV, 19-Jun-2020.) |
⊢ 𝑁 ∈ ℕ ⇒ ⊢ (𝑁 ∈ Even ↔ ¬ 𝑁 ∈ Odd ) | ||
Theorem | odd2np1ALTV 41585* | An integer is odd iff it is one plus twice another integer. (Contributed by Scott Fenton, 3-Apr-2014.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ ∃𝑛 ∈ ℤ ((2 · 𝑛) + 1) = 𝑁)) | ||
Theorem | oddm1evenALTV 41586 | An integer is odd iff its predecessor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 − 1) ∈ Even )) | ||
Theorem | oddp1evenALTV 41587 | An integer is odd iff its successor is even. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (𝑁 ∈ Odd ↔ (𝑁 + 1) ∈ Even )) | ||
Theorem | oexpnegALTV 41588 | The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.) (Revised by AV, 19-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ∧ 𝑁 ∈ Odd ) → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | ||
Theorem | oexpnegnz 41589 | The exponential of the negative of a number not being 0, when the exponent is odd. (Contributed by AV, 19-Jun-2020.) |
⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ Odd ) → (-𝐴↑𝑁) = -(𝐴↑𝑁)) | ||
Theorem | bits0ALTV 41590 | Value of the zeroth bit. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ ℤ → (0 ∈ (bits‘𝑁) ↔ 𝑁 ∈ Odd )) | ||
Theorem | bits0eALTV 41591 | The zeroth bit of an even number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ Even → ¬ 0 ∈ (bits‘𝑁)) | ||
Theorem | bits0oALTV 41592 | The zeroth bit of an odd number is zero. (Contributed by Mario Carneiro, 5-Sep-2016.) (Revised by AV, 19-Jun-2020.) |
⊢ (𝑁 ∈ Odd → 0 ∈ (bits‘𝑁)) | ||
Theorem | divgcdoddALTV 41593 | Either 𝐴 / (𝐴 gcd 𝐵) is odd or 𝐵 / (𝐴 gcd 𝐵) is odd. (Contributed by Scott Fenton, 19-Apr-2014.) (Revised by AV, 21-Jun-2020.) |
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → ((𝐴 / (𝐴 gcd 𝐵)) ∈ Odd ∨ (𝐵 / (𝐴 gcd 𝐵)) ∈ Odd )) | ||
Theorem | opoeALTV 41594 | The sum of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.) |
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴 + 𝐵) ∈ Even ) | ||
Theorem | opeoALTV 41595 | The sum of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.) |
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴 + 𝐵) ∈ Odd ) | ||
Theorem | omoeALTV 41596 | The difference of two odds is even. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.) |
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Odd ) → (𝐴 − 𝐵) ∈ Even ) | ||
Theorem | omeoALTV 41597 | The difference of an odd and an even is odd. (Contributed by Scott Fenton, 7-Apr-2014.) (Revised by AV, 20-Jun-2020.) |
⊢ ((𝐴 ∈ Odd ∧ 𝐵 ∈ Even ) → (𝐴 − 𝐵) ∈ Odd ) | ||
Theorem | oddprmALTV 41598 | A prime not equal to 2 is odd. (Contributed by Mario Carneiro, 4-Feb-2015.) (Revised by AV, 21-Jun-2020.) |
⊢ (𝑁 ∈ (ℙ ∖ {2}) → 𝑁 ∈ Odd ) | ||
Theorem | 0evenALTV 41599 | 0 is an even number. (Contributed by AV, 11-Feb-2020.) (Revised by AV, 17-Jun-2020.) |
⊢ 0 ∈ Even | ||
Theorem | 0noddALTV 41600 | 0 is not an odd number. (Contributed by AV, 3-Feb-2020.) (Revised by AV, 17-Jun-2020.) |
⊢ 0 ∉ Odd |
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