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In number theory, a narcissistic number [1] [2] (also known as a pluperfect digital invariant (PPDI), [3] an Armstrong number [4] (after Michael F. Armstrong) [5] or a plus perfect number) [6] in a given number base is a number that is the sum of its own digits each raised to the power of the number of digits.
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A number that is non-palindromic in all bases b in the range 2 ≤ b ≤ n − 2 can be called a strictly non-palindromic number. For example, the number 6 is written as "110" in base 2, "20" in base 3, and "12" in base 4, none of which are palindromes. All strictly non-palindromic numbers larger than 6 are prime.
These examples shouldn't have links, since they are base 3 and 4 numbers but the links are to base 10 numbers and so are meaningless --206.171.6.11 15:12, 8 November 2006 (UTC) Some base three Armstrong numbers are: 0,1,2,12,122; Some base four Armstrong numbers are: 0,1,2,3,313
There are finitely many natural numbers less than , so the number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point. For b = 2 {\displaystyle b=2} , the number of digits k ≤ n {\displaystyle k\leq n} for any number, once again, making it a preperiodic point.
The number Λ such that (,) = has real zeros if and only if λ ≥ Λ. where Φ ( u ) = ∑ n = 1 ∞ ( 2 π 2 n 4 e 9 u − 3 π n 2 e 5 u ) e − π n 2 e 4 u {\displaystyle \Phi (u)=\sum _{n=1}^{\infty }(2\pi ^{2}n^{4}e^{9u}-3\pi n^{2}e^{5u})e^{-\pi n^{2}e^{4u}}} .
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The naming procedure for large numbers is based on taking the number n occurring in 10 3n+3 (short scale) or 10 6n (long scale) and concatenating Latin roots for its units, tens, and hundreds place, together with the suffix -illion. In this way, numbers up to 10 3·999+3 = 10 3000 (short scale) or 10 6·999 = 10 5994 (long scale