`bigOmega_prime`

proved

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module: IndisputableMonolith.NumberTheory.Primes.ArithmeticFunctions
domain: NumberTheory
line: 458 · github
papers citing: none yet

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IndisputableMonolith.NumberTheory.Primes.ArithmeticFunctions on GitHub at line 458.

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formal source

 455  exact ArithmeticFunction.cardDistinctFactors_apply_prime hp'
 456
 457/-- Ω(p) = 1 for prime p. -/
 458theorem bigOmega_prime {p : ℕ} (hp : Prime p) : bigOmega p = 1 := by
 459  have hp' : Nat.Prime p := (prime_iff p).1 hp
 460  exact ArithmeticFunction.cardFactors_apply_prime hp'
 461
 462/-! ### Additional totient values -/
 463
 464/-- φ(2) = 1. -/
 465theorem totient_two : totient 2 = 1 := by
 466  simp [totient]
 467
 468/-- φ(3) = 2. -/
 469theorem totient_three : totient 3 = 2 := by
 470  native_decide
 471
 472/-- φ(4) = 2. -/
 473theorem totient_four : totient 4 = 2 := by
 474  native_decide
 475
 476/-- φ(5) = 4. -/
 477theorem totient_five : totient 5 = 4 := by
 478  native_decide
 479
 480/-- φ(6) = 2. -/
 481theorem totient_six : totient 6 = 2 := by
 482  native_decide
 483
 484/-- φ(8) = 4. -/
 485theorem totient_eight : totient 8 = 4 := by
 486  native_decide
 487
 488/-- φ(45) = 24. -/

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