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1: 24.10 Arithmetic Properties
where m n 0 ( mod p - 1 ) . …valid when m n ( mod ( p - 1 ) p ) and n 0 ( mod p - 1 ) , where ( 0 ) is a fixed integer. …
24.10.8 N 2 n 0 ( mod p ) ,
valid for fixed integers ( 1 ) , and for all n ( 1 ) such that 2 n 0 ( mod p - 1 ) and p | 2 n .
24.10.9 E 2 n { 0 ( mod p ) if  p 1 ( mod 4 ) , 2 ( mod p ) if  p 3 ( mod 4 ) ,
2: 27.16 Cryptography
Thus, y x r ( mod n ) and 1 y < n . … By the Euler–Fermat theorem (27.2.8), x ϕ ( n ) 1 ( mod n ) ; hence x t ϕ ( n ) 1 ( mod n ) . But y s x r s x 1 + t ϕ ( n ) x ( mod n ) , so y s is the same as x modulo n . …
3: 36.9 Integral Identities
§36.9 Integral Identities
36.9.9 | Ψ ( E ) ( x , y , z ) | 2 = 8 π 2 3 2 / 3 0 0 2 π ( Ai ( 1 3 1 / 3 ( x + i y + 2 z u exp ( i θ ) + 3 u 2 exp ( - 2 i θ ) ) ) Bi ( 1 3 1 / 3 ( x - i y + 2 z u exp ( - i θ ) + 3 u 2 exp ( 2 i θ ) ) ) ) u d u d θ .
4: 26.21 Tables
Andrews (1976) contains tables of the number of unrestricted partitions, partitions into odd parts, partitions into parts ± 2 ( mod 5 ) , partitions into parts ± 1 ( mod 5 ) , and unrestricted plane partitions up to 100. …
5: 22.9 Cyclic Identities
§22.9 Cyclic Identities
§22.9(ii) Typical Identities of Rank 2
§22.9(iii) Typical Identities of Rank 3
6: 24.5 Recurrence Relations
§24.5(ii) Other Identities
§24.5(iii) Inversion Formulas
In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa. …
7: 15.17 Mathematical Applications
§15.17(iv) Combinatorics
In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …
8: 27.15 Chinese Remainder Theorem
The Chinese remainder theorem states that a system of congruences x a 1 ( mod m 1 ) , , x a k ( mod m k ) , always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod m ), where m is the product of the moduli. …
9: 17.17 Physical Applications
In exactly solved models in statistical mechanics (Baxter (1981, 1982)) the methods and identities of §17.12 play a substantial role. …
10: 17.18 Methods of Computation
Lehner (1941) uses Method (2) in connection with the Rogers–Ramanujan identities. …