§ 3 Congruence

[ Congruence and its properties ]   Let m be a natural number , if the difference ab between integers a and b is a multiple of m , then a and b are said to be congruential modulo m , denoted as (mod m )

Otherwise record as (mod m )

Indicates that a and b are not congruent modulo m .

Congruence has the following properties :

1 ° (mod m ) ( reflexive ) 2 ° If (mod m ), then (mod m ) ( symmetry )  3 ° If , (mod m ), then (mod m ) ( transitive )   4 ° If , (mod m ), then   (mod m ) (mod m ) (mod m )

5 ° If , (mod m ), and , then (mod m ).    [ Complete Residual System and Reduced Residual System ]   Let m be the modulus , then by the congruence properties 1 ° , 2 ° , 3 ° , all integers can be divided into m classes , the numbers of the same class are congruent , and the numbers of different classes are different We call such a class a congruence class , and each class is represented by a number , for example form a complete residual system .

Take a representative from the classes that are relatively prime to m Constitute a reduced residual system ( referred to as a reduced system ), where j ( m) is the number of numbers that do not exceed m and are relatively prime to m ( called the Euler function ).

The residual system has the following properties :

1 ° If x passes through a complete residual system of m , and passes through a complete residual system , then passes through a complete residual system .     2 ° If x passes through m in a constricted system , and over a condensed line , then over a condensed line .     If 3 ° is a contraction system of the modulus m , and ( k, m )=1, then  is also a shortened system of m

[ Eulerian theorem ]   If ( k, m )=1, then (mod m )

where j (m) is the Euler function .

[ Fermat's theorem ]   If p is prime , then for all integers a , we have (mod p )

Obviously Fermat's theorem is a special case of Euler's theorem .

[ Conditions for Solvability of First-order Congruence Equations ]   Congruence Equations (mod m )

The necessary and sufficient conditions for a solution are : When this condition is satisfied , the number of solutions ( different remainders to modulo m ) is [ Solving method of first-order congruential equation in one variable ]   First- order congruential equation in one variable (mod m )                                  (1)

The necessary and sufficient conditions for a solution are : , if there is a solution, there are ( a , m ) mutually different complementary solutions , mod m . The solution is as follows :

Set ( a , m ) = d , , then the original equation (1) can be transformed into Referred to as (2)

First determined by rolling division ( see § 1), such that  but is the solution of the congruence equation (2) , and finally we get

x=x'+m't , t= is the solution of the original equation (1) , mod m . That is, there are d solutions Not congruent modulo m .

[ Sun Tzu's Theorem * ] If ij , then the system of congruence equations  (3)

There is a unique solution , . The solution of the system of congruence equations (3) is as follows :

Because , so it can be obtained by the tossing and dividing method , satisfying   remember then final calculation It is the only solution to the system of congruence equations (3) , . [ Quadratic residual and quadratic non-residual ]   Let m be an integer greater than 1 , ( n , m )=1, if (mod m )

solvable , then n is called the quadratic residue modulo m , or quadratic residue , mod m . Otherwise , n is called the quadratic non-residue modulo m .

The product of two quadratic residues is still quadratic residue , mod p.

The product of two quadratic non-residues is the quadratic residue , mod p.

The product of a quadratic residual and a quadratic non-residual is a quadratic non-residue , mod p.

Let p be an odd prime , then in the reduced residual system modulo p , there is a quadratic residual :  There is a quadratic non-residue , mod p . [ Legendre symbols and their properties ]   Let p be an odd prime number , and p n also set It is called Legendre notation . Legendre symbols have the following properties :

1 ° if (mod p ), p n then   = 2 ° Eulerian criterion : Let p be an odd prime number , then (mod p )

3 ° If ​​p is an odd prime number , p mn , then  4 ° If p is an odd prime number , then 5 ° If p is an odd prime number , then 6 ° Gaussian reciprocity law : Let p, q be two different odd prime numbers , then [ Number of solutions of quadratic congruence ]   Let l > 0, p n , if p is an odd prime number, then the quadratic congruence (mod ) The solution of . When p = 2 , there is a solution when l = 1 .  [ Number of times modulo m ] Let h be an integer , ( m , h )=1, satisfying (mod m )

The smallest positive integer l of is called the degree of h to modulo m , or the degree of h , mod m .

If (mod m ), then , where l is the number of h , mod m  [ Primitive roots and exponents of prime modulo ]   Let p be a prime number , and the number whose degree is p - 1 is called the primordial root modulo p .

Let g be a primitive root modulo p , then The two are not complementary to each other .

For any integer n ( p n ), there must be a number a such that (mod p )   , This a is called the exponent of n , mod p . It is written as When it is not easy to be confused , abbreviated ind n .

The index has the following properties :

1 ° if b is satisfied (mod p )

any number of , then (mod p - 1)

2 ° (mod p - 1), p ab  3 ° (mod p - 1), p a  4 ° The base interchange formula is set as two unequal primitive roots of mod p , and (mod p ), then   (mod p - 1)

[ Original root modulo m ] Let m be a natural number , if there is a number g such that (mod m )

If the two are not complementary to each other , then this g is called the primitive root of the modulo m, and the necessary and sufficient conditions for the existence of the primitive root of the modulo m are : ( p is an odd prime number , l is a positive integer )

[ List of prime numbers and their smallest primitive roots ( within 5000 )] The one with * means 10 is its primitive root

 p g p g p g p g 3 5 7 * 11 13 17 * 19 * 23 * 29 * 31 37 41 43 47 * 53 59 * 61 * 67 71 73 79 83 89 97 * 101 103 107 109 * 113 * 127 131 * 137 139 149 * 151 157 163 167 * 173 179 * 181 * 191 193 * 197 199 211 223 * 227 229 * 233 * 239 241 2 2 3 2 2 3 2 5 2 3 2 6 3 5 2 2 2 2 7 5 3 2 3 5 2 5 2 6 3 3 2 3 2 2 6 5 2 5 2 2 2 19 5 2 3 2 3 2 6 3 7 7 251 257 * 263 * 269 * 271 277 281 283 293 307 311 313 * 317 331 337 * 347 349 353 359 367 * 373 379 * 383 * 389 * 397 401 409 419 * 421 431 433 * 439 443 449 457 461 * 463 467 479 487 * 491 * 499 * 503 * 509 * 521 523 541 * 547 557 563 569 571 * 6 3 5 2 6 5 3 3 2 5 17 10 2 3 10 2 2 3 7 6 2 2 5 2 5 3 twenty one 2 2 7 5 15 2 3 13 2 3 2 13 3 2 7 5 2 3 2 2 2 2 2 3 3 577 * 587 593 * 599 601 607 613 617 619 * 631 641 643 647 * 653 659 * 661 673 677 683 691 701 * 709 * 719 727 * 733 739 743 * 751 757 761 769 773 787 797 809 811 * 821 * 823 * 827 829 839 853 857 * 859 863 * 877 881 883 887 * 907 911 919 5 2 3 7 7 3 2 3 2 3 3 11 5 2 2 2 5 2 5 3 2 2 11 5 6 3 5 3 2 6 11 2 2 2 3 3 2 3 2 2 11 2 3 2 5 2 3 2 5 2 17 7 929 937 * 941 * 947 953 * 967 971 * 977 * 983 * 991 997 1009 1013 1019 * 1021 * 1031 1033 * 1039 1049 1051 * 1061 1063 * 1069 * 1087 * 1091 * 1093 1097 * 1103 * 1109 * 1117 1123 1129 1151 1153 * 1163 1171 * 1181 * 1187 1193 * 1201 1213 * 1217 * 1223 * 1229 * 1231 1237 1249 1259 * 1277 1279 1283 1289 3 5 2 2 3 5 6 3 5 6 7 11 3 2 10 14 5 3 3 7 2 3 6 3 2 5 3 5 2 2 2 11 17 5 5 2 7 2 3 11 2 3 5 2 3 2 7 2 2 3 2 6

 p g p g p g p g 1291 * 1297 * 1301 * 1303 * 1307 1319 1321 1327 * 1361 1367 * 1373 1381 * 1399 1409 1423 1427 1429 * 1433 * 1439 1447 * 1451 1453 1459 1471 1481 1483 1487 * 1489 1493 1499 1511 1523 1531 * 1543 * 1549 * 1553 * 1559 1567 * 1571 * 1579 * 1583 * 1597 1601 1607 * 1609 1613 1619 * 1621 * 1627 1637 1657 1663 * p 2 10 2 6 2 13 13 3 3 5 2 2 13 3 3 2 6 3 7 3 2 2 5 6 3 2 5 14 2 2 11 2 2 5 2 3 19 3 2 3 5 11 3 5 7 3 2 2 3 2 11 3 g 1667 1669 1693 1697 * 1699 1709 * 1721 1723 1733 1741 * 1747 1753 1759 1777 * 1783 * 1787 1789 * 1801 1811 * 1823 * 1831 1847 * 1861 * 1867 1871 1873 * 1877 1879 1889 1901 1907 1913 * 1931 1933 1949 * 1951 1973 1979 * 1987 1993 * 1997 1999 2003 2011 2017 * 2027 2029 * 2039 2053 2063 * 2069 * 2081 p 2 2 2 3 3 3 3 3 2 2 2 7 6 5 10 2 6 11 6 5 3 5 2 2 14 10 2 6 3 2 2 3 2 5 2 3 2 2 2 5 2 3 5 3 5 2 2 7 2 5 2 3 g 2083 2087 2089 2099 * 2111 2113 * 2129 2131 2137 * 2141 * 2143 * 2153 * 2161 2179 * 2203 2207 * 2213 2221 * 2237 2239 2243 2251 * 2267 2269 * 2273 * 2281 2287 2293 2297 * 2309 * 2311 2333 2339 * 2341 * 2347 2351 2357 2371 * 2377 2381 2383 * 2389 * 2393 2399 2411 * 2417 * 2423 * 2437 * 2441 2447 * 2459 * 2467 p 2 5 7 2 7 5 3 2 10 2 3 3 twenty three 7 5 5 2 2 2 3 2 7 2 2 3 7 19 2 5 2 3 2 2 7 3 13 2 2 5 3 5 2 3 11 6 3 5 2 6 5 2 2 g 2473 * 2477 2503 2521 2531 2539 * 2543 * 2549 * 2551 2557 2579 * 2591 2593 * 2609 2617 * 2621 * 2633 * 2647 2657 * 2659 2663 * 2671 2677 2683 2687 * 2689 2693 2699 * 2707 2711 2713 * 2719 2729 * 2731 2741 * 2749 2753 * 2767 * 2777 * 2789 * 2791 2797 2801 2803 2819 * 2833 * 2837 2843 2851 * 2857 2861 * 2879 p 5 2 3 17 2 2 5 2 6 2 2 7 7 3 5 2 3 3 3 2 5 7 2 2 5 19 2 2 2 7 5 3 3 3 2 6 3 3 3 2 6 2 3 2 2 5 2 2 2 11 2 7 g 2887 2897 * 2903 * 2909 * 2917 2927 * 2939 * 2953 2957 2963 2969 2971 * 2999 3001 3011 * 3019 * 3023 * 3037 3041 3049 3061 3067 3079 3083 3089 3109 3119 3121 3137 * 3163 3167 * 3169 3181 3187 3191 3203 3209 3217 3221 * 3229 3251 * 3253 3257 * 3259 * 3271 3299 * 3301 * 3307 3313 * 3319 3323 3329 3331 * p 5 3 5 2 5 5 2 13 2 2 3 10 17 14 2 2 5 2 3 11 6 2 6 2 3 6 7 7 3 3 5 7 7 2 11 2 3 5 10 6 6 2 3 3 3 2 6 2 10 6 2 3 3 g 3343 * 3347 3359 3361 3371 * 3373 3389 * 3391 3407 * 3413 3433 * 3449 3457 3461 * 3463 * 3467 3469 * 3491 3499 3511 3517 3527 * 3529 3533 3539 * 3541 3547 3557 3559 3571 * 3581 * 3583 3593 * 3607 * 3613 3617 * 3623 * 3631 3637 3643 3659 * 3671 3673 * 3677 3691 3697 3701 * 3709 * 3719 3727 * 3733 3739 3761 p 5 2 11 twenty two 2 5 3 3 5 2 5 3 7 2 3 2 2 2 2 7 2 5 17 2 2 7 2 2 3 2 2 3 3 5 2 3 5 twenty one 2 2 2 13 5 2 2 5 2 2 7 3 2 7 3 g 3767 3769 3779 * 3793 3797 3803 3821 * 3823 3833 * 3847 * 3851 * 3853 3863 * 3877 3881 3889 3907 3911 3917 3919 3923 3929 3931 3943 * 3947 3967 * 3989 * 4001 4003 4007 * 4013 4019 * 4021 4027 4049 4051 * 4057 * 4073 * 4079 4091 * 4093 4099 4111 4127 4129 4133 4139 * 4153 * 4157 4159 4177 * 4201 4211 * p 5 7 2 5 2 2 3 3 3 5 2 2 5 2 13 11 2 13 2 3 2 3 2 3 2 6 2 3 2 5 2 2 2 3 3 10 5 3 11 2 2 2 17 5 13 2 2 5 2 3 5 11 6 g 4217 * 4219 * 4229 * 4231 4241 4243 4253 4259 * 4261 * 4271 4273 4283 4289 4297 4327 * 4337 * 4339 * 4349 * 4357 4363 4373 4391 4397 4409 4421 * 4423 * 4441 4447 * 4451 * 4457 * 4463 * 4481 4483 4493 4507 4513 4517 4519 4523 4547 4549 4561 4567 * 4583 * 4591 4597 4603 4621 4637 4639 4643 4649 4651 * p 3 2 2 3 3 2 2 2 2 7 5 2 3 5 3 3 10 2 2 2 2 14 2 3 3 3 twenty one 3 2 3 5 3 2 2 2 7 2 3 5 2 6 11 3 5 11 5 2 2 2 3 5 3 3 g 4657 4663 4673 * 4679 4691 * 4703 * 4721 4723 4729 4733 15 3 3 11 2 5 6 2 17 5 4751 4759 4783 * 4787 4789 4793 * 4799 4801 4813 4817 * 19 3 6 2 2 3 7 7 2 3 4831 4861 4871 4877 4889 4903 4909 4919 4931 * 4933 3 11 11 2 3 3 6 13 6 2 4937 * 4943 * 4951 4957 4967 * 4969 4973 4987 4993 4999 3 7 6 2 5 11 2 2 5 3

* Foreign books and periodicals call Sun Tzu's theorem the Chinese remainder theorem. In ancient my country, the solution to this kind of problem has been described in the question "I don't know how many things" in "Sun Tzu's Suanjing".