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1: 27.15 Chinese Remainder Theorem
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►Their product has 20 digits, twice the number of digits in the data.
…These numbers, in turn, are combined by the Chinese remainder theorem to obtain the final result , which is correct to 20 digits.
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2: 24.20 Tables
3: 26.14 Permutations: Order Notation
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►As an example, is an element of The inversion number is the number of pairs of elements for which the larger element precedes the smaller:
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►The Eulerian number, denoted , is the number of permutations in with exactly descents.
…The Eulerian number
is equal to the number of permutations in with exactly excedances.
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§26.14(iii) Identities
…4: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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5: Bibliography L
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An asymptotic estimate for the Bernoulli and Euler numbers.
Canad. Math. Bull. 20 (1), pp. 109–111.
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6: 26.13 Permutations: Cycle Notation
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►The Stirling cycle numbers of the first kind, denoted by , count the number of permutations of with exactly cycles.
They are related to Stirling numbers of the first kind by
…See §26.8 for generating functions, recurrence relations, identities, and asymptotic approximations.
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►The derangement number, , is the number of elements of with no fixed points:
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►A permutation is even or odd according to the parity of the number of transpositions.
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7: 25.6 Integer Arguments
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25.6.3
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8: 27.2 Functions
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►where are the distinct prime factors of , each exponent is positive, and is the number of distinct primes dividing .
…Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
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