NICE [Titu_Andreescu,_Dorin_Andrica]_Number_theory_Str(BookZZ.org)

GLOSSARY 399

Pythagorean equation
The Diophantine equation x2 + y2 = z2.

Pythagorean triple
A triple of the form (m2 − n2, 2mn, m2 + n2), where m and n are positive
integers.

Quadratic residue mod m
Let a and m be positive integers such that gcd(a, m) = 1. We say that a
is a quadratic residue mod m if the convergence x2 ≡ a (mod m) has a
solution.

Quadratic Reciprocity Law of Gauss
If p and q are distinct odd primes, then

q p = (−1) p−1 · q−1
2 2
pq

Sum of divisors
For a positive integer n denote by σ(n) the sum of its positive divisors
including 1 and n itself. It is clear that

σ(n) = d

d|n

Summation function
For an arithmetic function f the function F defined by

F (n) = f (d)

d|n

Wilson’s Theorem
For any prime p, (p − 1)! ≡ −1 (mod p).

400 GLOSSARY

References

[1] Andreescu, T.; Feng, Z., 101 Problems in Algebra from the Training
of the USA IMO Team, Australian Mathematics Trust, 2001.

[2] Andreescu, T.; Feng, Z., 102 Combinatorial Problems from the Train-
ing of the USA IMO Team, Birkha¨user, 2002.

[3] Andreescu, T.; Feng, Z., 103 Trigonometry Problems from the Training
of the USA IMO Team, Birkha¨user, 2004.

[4] Feng, Z.; Rousseau, C.; Wood, M., USA and International Mathemat-
ical Olympiads 2005, Mathematical Association of America, 2006.

[5] Andreescu, T.; Feng, Z.; Loh, P., USA and International Mathematical
Olympiads 2004, Mathematical Association of America, 2005.

[6] Andreescu, T.; Feng, Z., USA and International Mathematical
Olympiads 2003, Mathematical Association of America, 2004.

[7] Andreescu, T.; Feng, Z., USA and International Mathematical
Olympiads 2002, Mathematical Association of America, 2003.

[8] Andreescu, T.; Feng, Z., USA and International Mathematical
Olympiads 2001, Mathematical Association of America, 2002.

402 References

[9] Andreescu, T.; Feng, Z., USA and International Mathematical
Olympiads 2000, Mathematical Association of America, 2001.

[10] Andreescu, T.; Feng, Z.; Lee, G.; Loh, P., Mathematical Olympiads:
Problems and Solutions from around the World, 2001–2002, Mathe-
matical Association of America, 2004.

[11] Andreescu, T.; Feng, Z.; Lee, G., Mathematical Olympiads: Problems
and Solutions from around the World, 2000–2001, Mathematical As-
sociation of America, 2003.

[12] Andreescu, T.; Feng, Z., Mathematical Olympiads: Problems and So-
lutions from around the World, 1999–2000, Mathematical Association
of America, 2002.

[13] Andreescu, T.; Feng, Z., Mathematical Olympiads: Problems and So-
lutions from around the World, 1998–1999, Mathematical Association
of America, 2000.

[14] Andreescu, T.; Kedlaya, K., Mathematical Contests 1997–1998:
Olympiad Problems from around the World, with Solutions, American
Mathematics Competitions, 1999.

[15] Andreescu, T.; Kedlaya, K., Mathematical Contests 1996–1997:
Olympiad Problems from around the World, with Solutions, American
Mathematics Competitions, 1998.

[16] Andreescu, T.; Kedlaya, K.; Zeitz, P., Mathematical Contests 1995–
1996: Olympiad Problems from around the World, with Solutions,
American Mathematics Competitions, 1997.

[17] Andreescu, T.; Enescu, B., Mathematical Olympiad Treasures,
Birkh¨auser, 2003.

[18] Andreescu, T.; Gelca, R., Mathematical Olympiad Challenges,
Birkh¨auser, 2000.

[19] Andreescu, T., Andrica, D., An Introduction to Diophantine Equa-
tions, GIL Publishing House, 2002.

[20] Andreescu, T.; Andrica, D., 360 Problems for Mathematical Contests,
GIL Publishing House, 2003.

References 403

[21] Andreescu, T.; Andrica, D.; Feng, Z., 104 Number Theory Problems
From the Training of the USA IMO Team, Birkh¨auser, 2006 (to ap-
pear).

[22] Djuiki´c, D.; Jankovi´c, V.; Mati´c, I.; Petrovi´c, N., The IMO Com-
pendium, A Collection of Problems Suggested for the International
Mathematical Olympiads: 1959-2004, Springer, 2006.

[23] Doob, M., The Canadian Mathematical Olympiad 1969–1993, Univer-
sity of Toronto Press, 1993.

[24] Engel, A., Problem-Solving Strategies, Problem Books in Mathematics,
Springer, 1998.

[25] Everest, G., Ward, T., An Introduction to Number Theory, Springer,
2005.

[26] Fomin, D.; Kirichenko, A., Leningrad Mathematical Olympiads 1987–
1991, MathPro Press, 1994.

[27] Fomin, D.; Genkin, S.; Itenberg, I., Mathematical Circles, American
Mathematical Society, 1996.

[28] Graham, R. L.; Knuth, D. E.; Patashnik, O., Concrete Mathematics,
Addison-Wesley, 1989.

[29] Gillman, R., A Friendly Mathematics Competition, The Mathematical
Association of America, 2003.

[30] Greitzer, S. L., International Mathematical Olympiads, 1959–1977,
New Mathematical Library, Vol. 27, Mathematical Association of
America, 1978.

[31] Grosswald, E., Topics from the Theory of Numbers, Second Edition,
Birkha¨user, 1984.

[32] Hardy, G.H.; Wright, E.M., An Introduction to the Theory of Numbers,
Oxford University Press, 5th Edition, 1980.

[33] Holton, D., Let’s Solve Some Math Problems, A Canadian Mathemat-
ics Competition Publication, 1993.

404 References

[34] Kedlaya, K; Poonen, B.; Vakil, R., The William Lowell Putnam Math-
ematical Competition 1985–2000, The Mathematical Association of
America, 2002.

[35] Klamkin, M., International Mathematical Olympiads, 1978–1985, New
Mathematical Library, Vol. 31, Mathematical Association of America,
1986.

[36] Klamkin, M., USA Mathematical Olympiads, 1972–1986, New Mathe-
matical Library, Vol. 33, Mathematical Association of America, 1988.

[37] Ku¨rsch´ak, J., Hungarian Problem Book, volumes I & II, New Mathe-
matical Library, Vols. 11 & 12, Mathematical Association of America,
1967.

[38] Kuczma, M., 144 Problems of the Austrian–Polish Mathematics Com-
petition 1978–1993, The Academic Distribution Center, 1994.

[39] Kuczma, M., International Mathematical Olympiads 1986–1999,
Mathematical Association of America, 2003.

[40] Larson, L. C., Problem-Solving Through Problems, Springer-Verlag,
1983.

[41] Lausch, H. The Asian Pacific Mathematics Olympiad 1989–1993, Aus-
tralian Mathematics Trust, 1994.

[42] Liu, A., Chinese Mathematics Competitions and Olympiads 1981–
1993, Australian Mathematics Trust, 1998.

[43] Liu, A., Hungarian Problem Book III, New Mathematical Library, Vol.
42, Mathematical Association of America, 2001.

[44] Lozansky, E.; Rousseau, C. Winning Solutions, Springer, 1996.

[45] Mordell, L.J., Diophantine Equations, Academic Press, London and
New York, 1969.

[46] Niven, I., Zuckerman, H.S., Montgomery, H.L., An Introduction to the
Theory of Numbers, Fifth Edition, John Wiley & Sons, Inc., New York,
Chichester, Brisbane, Toronto, Singapore, 1991.

References 405

[47] Savchev, S.; Andreescu, T. Mathematical Miniatures, Anneli Lax New
Mathematical Library, Vol. 43, Mathematical Association of America,
2002.

[48] Shklarsky, D. O; Chentzov, N. N; Yaglom, I. M., The USSR Olympiad
Problem Book, Freeman, 1962.

[49] Slinko, A., USSR Mathematical Olympiads 1989–1992, Australian
Mathematics Trust, 1997.

[50] Szekely, G. J., Contests in Higher Mathematics, Springer-Verlag, 1996.

[51] Tattersall, J.J., Elementary Number Theory in Nine Chapters, Cam-
bridge University Press, 1999.

[52] Taylor, P. J., Tournament of Towns 1980–1984, Australian Mathe-
matics Trust, 1993.

[53] Taylor, P. J., Tournament of Towns 1984–1989, Australian Mathe-
matics Trust, 1992.

[54] Taylor, P. J., Tournament of Towns 1989–1993, Australian Mathe-
matics Trust, 1994.

[55] Taylor, P. J.; Storozhev, A., Tournament of Towns 1993–1997, Aus-
tralian Mathematics Trust, 1998.

406 References

Index of Authors

408 Index of Authors

Subject Index

arithmetic function, 119 Euler’s criterion, 180
Euler’s Theorem, 74, 147
base b representation, 51 Euler’s totient function, 132
Binet’s formula, 193
binomial coefficients, 211 Fermat’s Little Theorem, 141
Bonse’s inequality, 30 Fermat’s numbers, 189
Fibonacci numbers, 194
canonical factorization, 23 floor, 77
Carmichael’s integers, 142 fractional part, 77
characteristic equation, 198 fully divides, 25
Chinese Remainder Theorem, 47
composite, 22 Giuga’s conjecture, 267
congruence relation, 42 greatest common divisor, 30
congruent modulo n, 42
convolution inverse, 123 Inclusion-Exclusion Principle,
convolution product, 122 115
cubic equations, 171
infinite descent, 113
decimal representation, 51
Division Algorithm, 15 Kronecker’s theorem, 259

Euclidean Algorithm, 31 lattice point, 48
least common multiple, 32
Legendre’s formula, 136

410 Subject Index squarefree, 61
sum of divisors, 129
Legendre’s function, 136 sum of the digits, 94
Legendre’s symbol, 179 summation function, 120
linear Diophantine equation, 157
linear recurrence of order k, 197 twin primes, 25
Lucas’ sequence, 195
Vandermonde property, 212
M¨obius function, 120
mathematical induction, 108 Wilson’s Theorem, 153
Mersenne’s numbers, 191

Niven number, 288
number of divisors, 126

order of a modulo n, 150

Pascal’s triangle, 211
Pell’s equation, 164
Pell’s sequence, 199
perfect cube, 70
perfect numbers, 192
perfect power, 61
perfect square, 61
Pigeonhole Principle, 106
prime, 21
prime factorization theorem, 22
Prime Number Theorem, 24
primitive root modulo n, 150
primitive solution, 161
problem of Frobenius, 336
Pythagorean equation, 161
Pythagorean triple, 162

Quadratic Reciprocity Law of
Gauss, 182

quadratic residue, 179
quotient, 16

relatively prime, 30
remainder, 16