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 (= 13 + 103) = 509 + 6E4 (= 93 + X3), and it is also the smallest absolute Euler pseudoprime. (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.
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 number.
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 lowest number which can be represented by a Loeschian quadratic form a² + ab + b² 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).
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 103 ≡ −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).