Pandigital number

The first few pandigital base 10 numbers are : : 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689 The smallest pandigital number in a given base b is an integer of the form : b^{b - 1} + \sum_{d = 2}^{b - 1} db^{b - 1 - d} = \frac{b^b - b}{(b-1)^2} + (b-1) \times b^{b-2} - 1 The following table lists the smallest pandigital numbers of a few selected bases: As well as some other numeral systems: gives the base 10 values for the first 18 bases. In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form 2^n - 1 (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of b^x consecutive pandigital numbers with redundant digits by writing all the digits of the base together (but not putting the zero first as the most significant digit) and adding x + 1 zeroes at the end as least significant digits. Conversely, the smaller the base, b, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10. The formula is (b-1)(b-1)!. ==Variants and properties==

Sometimes, the term is used to refer only to pandigital numbers with no redundant digits. In some cases, a number might be called pandigital even if it does not have a zero as a significant digit, for example, 923456781 (these are also referred to as "penholodigital" or "zeroless pandigital numbers"). No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45, passing the divisibility rule for both 3 and 9. The first base 10 pandigital prime is 10123457689; lists more. For reasons of symmetry, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 1023456789876543201. The largest pandigital number without redundant digits to be also a square number is 9814072356 = 990662. Two of the zeroless pandigital Friedman numbers are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34. A pandigital Friedman number without redundant digits is the square: 2170348569 = 465872 + (0 × 139). The concept of a "pandigital approximation" was introduced by Erich Friedman in 2004. With the digits from 1 to 9 (each used exactly once) and the mathematical symbols + − × / ( ) . and ^, Euler's number can be approximated as (1+9^{-4^{7\times 6}})^{3^{2^{85}}}, which is correct to 1.8\cdot10^{25} decimal places. The variant (1+.2^{9^{7\times 6}})^{5^{3^{84}}} produces 8.3\cdot10^{39} correct digits. While much of what has been said does not apply to Roman numerals, there are pandigital numbers: , , , , , , , . These, listed in , use each of the characters just once, while has pandigital Roman numerals with repeats. Pandigital numbers are useful in fiction and in advertising. The Social Security number 987-65-4321 is a zeroless pandigital number reserved for use in advertising. Some credit card companies use pandigital numbers with redundant digits as fictitious credit card numbers (while others use strings of zeroes). == Arithmetic properties ==

Several arithmetic properties of pandigital numbers have been studied, in particular regarding divisibility and prime factorization. There are 244,423 cases in which a pandigital number is an integer multiple of another pandigital number. These frequently form multiplication chains consisting of several successive steps, with identical or different factors. Among these, 6,760 pandigital numbers initiate a chain with at least two steps, and 68 initiate a chain with three steps. The smallest prime number that is not a divisor of any pandigital number is 111,119. However, some larger prime numbers do occur as divisors of pandigital numbers. The largest prime divisor of a pandigital number is 1,097,393,447; multiplied by 9, it yields the pandigital number 9,876,541,023. In total, there are 87 pandigital square numbers. No pandigital cube and no pandigital number with an integer root of higher degree are known. Only three pandigital numbers have prime factorizations consisting exclusively of powers of one-digit primes (2, 3, 5, and 7): • 3.891.240.675 = 33 * 52 * 78 • 6.195.732.480 = 213 * 32 * 5 * 75 • 9.487.215.360 = 28 * 32 * 5 * 77 For each natural number n, there exists a maximum exponent k such that nk divides at least one pandigital number, while nk+1 does not divide any. For example, three pandigital numbers are divisible by 221, but none by 222: • 3.076.521.984 = 221 * 32 * 163 • 3.718.250.496 = 221 * 32 * 197 • 6.398.410.752 = 221 * 33 * 113 With respect to 3, only one pandigital number is divisible by 315, and none by 316: • 7.246.198.035 = 315 * 5 * 101 Analogously, only one pandigital number is divisible by 59, while 510 does not occur: • 9.123.046.875 = 33 * 59 * 173 This behavior extends to other integers. For example, 13 pandigital numbers are divisible by 68, whereas 69 does not occur as a divisor of any pandigital number. For 7, the highest power is 78 , represented by: • 3.891.240.675 = 33 * 52 * 78 • Beside 7, this one consists only of powers of 3 and 5. The three named cases divisible by 221 also represent 410 and 87 as the highest power concerning 4 and 8 which are powers of 2. Nine pandigital numbers are divisible by 97 = 314, among them the named one divisible by 315 ==Examples in base 10==

• 123456789 = The first zeroless pandigital number. • 381654729 = The only zeroless pandigital number where the first digits are divisible by . • 987654321 = The largest zeroless pandigital number without redundant digits. • 1023456789 = The first pandigital number. • 1234567890 = The pandigital number with the digits in order. • 3816547290 = The polydivisible pandigital number; the only pandigital number where the first digits are divisible by . • 9814072356 = The largest pandigital square without redundant digits. It is the square of 99066. • 9876543210 = The largest pandigital number without redundant digits. • 12345678987654321 = A pandigital number with all the digits except zero in both ascending and descending order. It is the square of 111111111; see Demlo number. It is also a palindromic number. ==See also==