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1: 24.10 Arithmetic Properties
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►where .
…valid when and , where is a fixed integer.
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24.10.8
►valid for fixed integers , and for all such that
and .
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24.10.9
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2: 25.17 Physical Applications
3: 26.21 Tables
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►Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts , partitions into parts , and unrestricted plane partitions up to 100.
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4: 27.16 Cryptography
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►Thus, and .
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►By the Euler–Fermat theorem (27.2.8), ; hence .
But , so is the same as modulo .
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5: 27.15 Chinese Remainder Theorem
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►The Chinese remainder theorem states that a system of congruences , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod ), where is the product of the moduli.
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6: 27.8 Dirichlet Characters
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27.8.6
►A Dirichlet character is called primitive (mod ) if for every proper divisor of (that is, a divisor ), there exists an integer , with and .
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27.8.7
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7: 24.15 Related Sequences of Numbers
8: 27.11 Asymptotic Formulas: Partial Sums
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27.11.9
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27.11.11
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►Letting in (27.11.9) or in (27.11.11) we see that there are infinitely many primes if are coprime; this is Dirichlet’s theorem
on primes in arithmetic progressions.
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►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3).
The prime number theorem for
arithmetic progressions—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if , then the number of primes with is asymptotic to as .
9: 27.19 Methods of Computation: Factorization
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►Type II probabilistic algorithms for factoring rely on finding a pseudo-random pair of integers that satisfy .
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10: 26.2 Basic Definitions
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►Given a finite set with permutation , a cycle is an ordered equivalence class of elements of where is equivalent to if there exists an such that , where and is the composition of with .
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