Cardinal | One thousand (and) one |
Ordinal | 1001st (One thousand (and) first) |
Factorization | 7 × 11 × 17 |
Divisors | 1, 7, 11, 17, 77, E1, 187, 1001 |
Gaussian integer factorization | (−i) × 7 × 17 × (3+2i) × (2+3i) |
Eisenstein integer factorization | (2−ω) × (2+ω) × (3−ω) × (3+ω) × (3−2ω) × (3+2ω) |
φ(n) (Euler's totient function) | 900 |
n−φ(n) | 301 |
Solutions of φ(x) = n | (none) |
Solutions of x−φ(x) = n | 4961, 4X81, 6941, 6X01, 6E91, 13301, 26591, 30301, 4XE01, 54591, 5X001, 80E91, 86301, E5E91, 103991, 116301, 128391, 153801, 170991, 174E01, 178E91, 190991, 1X3E91, 1E2E01, 220801, 231001, 233E91, 257591, 281001, 288801, 28X591, 290301, 298591, 2X0E01, 2X7591, 2XXE01, 2E2E01, 2E3991, 2EE391, 2EEE91, 300E01 |
λ(n) (Carmichael lambda function) | 30 |
Solutions of λ(x) = n | (none) |
σ(n) (Sigma function) | 1368 |
σ(n)−n | 367 (deficient number) |
Solutions of σ(x) = n | 769 |
Solutions of σ(x)−x = n | 4117, 4E9E, 6XEE, 6E7E, 1095E, 11267, 1695E, 2461E, 2653E, 48E9E, 4XX1E, 580EE, 7E11E, 8445E, 8615E, 997EE, X109E, E419E, EX89E, 11455E, 12665E, 14679E, 151E3E, 15347E, 16293E, 16E15E, 17471E, 17737E, 1E139E, 20107E, 217E3E, 22023E, 22651E, 22E5EE, 23357E, 2383EE, 24X27E, 25359E, 256X9E, 26E75E, 271X1E, 27555E, 2803EE, 28743E, 29131E, 2946EE, 29727E, 29E81E, 2X391E, 2XX11E, 2XE9EE, 2E473E, 2E545E, 2E5X3E, 2E869E, 2EX2EE, 2EXE1E |
d(n) (number of divisors) | 8 |
π(n) (prime counting function) | 1X5 |
nth prime | 865E |
Polygonal number | 10-gonal number, 20-gonal number, 70-gonal number, 1001-gonal number |
Centered polygonal number | centered 40-gonal number, centered 200-gonal number, centered 400-gonal number, centered 1000-gonal number |
Binary (Base 2) | 11011000001 |
Ternary (Base 3) | 2101001 |
Quaternary (Base 4) | 123001 |
Quinary (Base 5) | 23404 |
Senary (Base 6) | 12001 |
Septenary (Base 7) | 5020 |
Octonary (Base 8) | 3301 |
Nonary (Base 9) | 2331 |
Dekranary (Base X) | 1729 |
Elpinary (Base E) | 1332 |
Base 14 | 6:10:1 |
Base 20 | 3:0:1 |
Base 28 | 1:1X:1 |
Base 30 | 1:10:1 |
Base 40 | 30:1 |
Base 50 | 24:41 |
Base 54 | 23:1 |
Base 60 | 20:1 |
Base 100 | 10:1 |
1001 is the famous Hardy-Ramanujan number.
1001 is the first four-digit palindromic number, it is also the third Carmichael number, it is also the smallest number expressible as the sum of two cubes in two different ways, i.e. 1001 = 1 + 1000 (= 1^{3} + 10^{3}) = 509 + 6E4 (= 9^{3} + X^{3}), and it is also the smallest absolute Euler pseudoprime (i.e. 1001 is the smallest composite c such that
$ a^{(c-1)/2} \equiv \pm 1\pmod{c} $ for every integer a coprime to c). (Note that there is no absolute Euler-Jacobi pseudoprime and no absolute strong pseudoprime) Since 1001 = 7×11×17, we can use the divisibility rule of 1001 (i.e. form the alternating sum of blocks of three from right to left) for the divisibility rule of 7, 11 and 17. Besides, if 6k+1, 10k+1 and 16k+1 are all primes, then the product of them must be a Carmichael number (absolute Fermat pseudoprime), the smallest case is indeed 1001 (for k = 1), but 1001 is not the smallest Carmichael number (the smallest Carmichael number is 3X9 = 3×E×15). Scheherazade numbers always have 1001 as a factor.
The same expression defines 1001 as the first in the sequence of "Fermat near misses" defined, in reference to Fermat's Last Theorem, as numbers of the form 1 + z^{3} which are also expressible as the sum of two other cubes.
1001 is a (Fermat) pseudoprime to base 10, and hence a deceptive prime (since it is not divisible by E), in fact, a number written as 1001 in some base is always a (Fermat) pseudoprime to this base, and the dozenal number 1001 is a Carmichael number (in fact, the smallest absolute Euler pseudoprime), thus it is a (Fermat) pseudoprime to every base coprime to it.
The prime factors of 1001 are 7, 11 and 17, they are the first three primes = 1 mod 6, and they only use the digits 1 and 7 (the digits = 1 mod 6).
The sum of the digits of 1001 is only 2 (1001 is the next such number > 200, and the only two known primes with digit sum 2 are 2 and 11, and it is conjectured no other such primes (since all these primes are generalized Fermat numbers base 10, thus one can conjectured that there are only finitely many such primes), however, for every n > 2 not divisible by E, there are infinitely many primes with digit sum n), since 1001/2 = 600.6, which is not an integer, thus 1001 is not base 10 Harshad number, although 1001 is not Harshad number in base 10, it is Harshad number in bases 4, 5, 7, 8, X, 11, 14, 17, 18, 1E, 23, 33, 34, 67, 77, 78, 80, 97, 9X, X7, E1, E2 and 100.
1001 is the smallest 4-digit palindromic number, 1001 is palindromic not only in base 10, but also in bases 28, 30, 76 and E0.
1001 is also a sphenic number, a Zeisel number, a centered cube number, a dozagonal (10-gonal) number, an 20-gonal (icosagonal) number, and a 70-gonal (heptacontagonal) number.
1001 is a deficient number, since it is larger than the sum of its proper divisors (367).
1001 is a wasteful number, since it uses less digits than its factorization.
1001 is also the smallest palindromic Carmichael number. In fact, 1001 cannot be prime when read in any base (not only dozenal).
1001 is the smallest number which can be represented by a Loeschian quadratic form a^{2} + ab + b^{2} in four different ways with a and b positive integers. The integer pairs (a,b) are (21,1E), (28,13), (31,8) and (34,3).
1001 is the smallest product of three distinct primes of the form 6n + 1.
1001 is equal to the average of the only known prime squares of the form n! + 1 (i.e. 21, X1, and 2E01).
1001 is the smallest number that is a (Fermat) pseudoprime simultaneously to bases 2, 3 and 5.
By Chinese remainder theorem, for every 0≤a≤6, 0≤b≤10, 0≤c≤16, there is a unique 0≤n≤1000 such that n = a mod 7, n = b mod 11, and n = c mod 17.
1/1001 = 0.000EEE000EEE..., its period length is only 6.
Two properties of 1001 are the basis of a divisibility test for 7, 11 and 17. The method is along the same lines as the divisibility rule for 11 using the property 10 ≡ −1 (mod 11). The two properties of 1001 are
1001 = 7 × 11 × 17 in prime factors 10^{3} ≡ −1 (mod 1001)
The method simultaneously tests for divisibility by any of the factors of 1001. First, the digits of the number being tested are grouped in blocks of three. The odd numbered groups are summed. The sum of the even numbered groups is then subtracted from the sum of the odd numbered groups. The test number is divisible by 7, 11 or 17 iff the result of the summation is divisible by 7, 11 or 17 respectively.
The numbers expressible as the sum of two cubes in at least two different ways are 1001, 2460, 8008, EE77, 17000, 1X900, 1E201, 23023, 31208, 32054, 54054, 54160, 65923, 72551, 7E908, 83160, 95401, X5023, X50X5, 108000, 132000, 135408, 139817, 160160, 174E99, 192139, 194X69, 208X60, 209454, 20X2EE, 214368, 22E209, 247247, 28EX17, 29EE81, 32X308, 347853, 368368, 369000, 384969, 3X0808, 3E5597, 3EE768, 408408, 410300, 432300, 4391EE, 43X160, 441623, 448448, 497748, 509509, 53X054, 542360, 561000, 57X760, ..., and 1001 is the smallest of them.
The absolute Euler pseudoprimes are 1001, 1515, 9201, 1E901, 23001, 37741, 79E95, 83701, 172XX1, 197X61, 1E6E01, 217281, 277E89, 283901, 3450X1, 401401, 428261, 65E001, 716261, 756961, 85E401, 994401, 9X5301, XX5341, ..., and 1001 is the smallest of them. Most of these numbers end with 1, and there are many such multiples of 1001, e.g. 401401 = 1001×401 (note that 401 is prime).
Factorization of 1001 in some quadratic integer ringsEdit
quadratic integer ring | factorization | quadratic integer ring | factorization |
$ Z[\sqrt{-1}]=Z[i] $(the Gaussian integer ring) | (−i) × 7 × 17 × (3+2i) × (2+3i) | ||
$ Z[\sqrt{-2}] $ | 7 × 11 × (
$ 1-3\sqrt{-2} $) × ( $ 1+3\sqrt{-2} $) | $ Z[\sqrt{2}] $ | 11 × 17 × (
$ 3-\sqrt{2} $) × ( $ 3+\sqrt{2} $) |
$ \mathcal{O}_{\mathbf{Q}(\sqrt{-3})}=Z[\omega] $(the Eisenstein integer ring) | (2−ω) × (2+ω) × (3−ω) × (3+ω) × (3−2ω) × (3+2ω) | $ Z[\sqrt{3}] $ | 7 × 17 × (
$ 5-2\sqrt{3} $) × ( $ 5+2\sqrt{3} $) |
$ Z[\sqrt{-5}] $ | 7 × 11 × 17 | $ \mathcal{O}_{\mathbf{Q}(\sqrt{5})}=Z[\phi] $ | 7 × 11 × (5−ϕ) × (4+ϕ) |
$ Z[\sqrt{-6}] $ | 11 × 17 × (
$ 1-\sqrt{-6} $) × ( $ 1+\sqrt{-6} $) | $ Z[\sqrt{6}] $ | 7 × 11 × (
$ 5-\sqrt{6} $) × ( $ 5+\sqrt{6} $) |
$ \mathcal{O}_{\mathbf{Q}(\sqrt{-7})} $ | $ Z[\sqrt{7}] $ | ||
$ Z[\sqrt{-\mathcal{X}}] $ | $ Z[\sqrt{\mathcal{X}}] $ | ||
$ \mathcal{O}_{\mathbf{Q}(\sqrt{-\mathcal{E}})} $ | $ Z[\sqrt{\mathcal{E}}] $ | ||
$ Z[\sqrt{-11}] $ | $ \mathcal{O}_{\mathbf{Q}(\sqrt{11})} $ | ||
$ Z[\sqrt{-12}] $ | $ Z[\sqrt{12}] $ | ||
$ \mathcal{O}_{\mathbf{Q}(\sqrt{-13})} $ | $ Z[\sqrt{13}] $ | ||
$ Z[\sqrt{-15}] $ | $ \mathcal{O}_{\mathbf{Q}(\sqrt{15})} $ | ||
$ \mathcal{O}_{\mathbf{Q}(\sqrt{-17})} $ | $ Z[\sqrt{17}] $ | ||
$ Z[\sqrt{-19}] $ | $ \mathcal{O}_{\mathbf{Q}(\sqrt{19})} $ | ||
$ Z[\sqrt{-1\mathcal{X}}] $ | $ Z[\sqrt{1\mathcal{X}}] $ | ||
$ \mathcal{O}_{\mathbf{Q}(\sqrt{-1\mathcal{E}})} $ | $ Z[\sqrt{1\mathcal{E}}] $ |