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Number Theory: Results and Theorems from Math 104A, Fall 2002, Study notes of Number Theory

University of California - San DiegoNumber Theory

The results and theorems proved in a number theory course, including formulas for sums of integers, binomial coefficients, divisibility, the floor function, prime numbers, greatest common divisors, the fundamental theorem of arithmetic, diophantine equations, congruences, inverses modulo m, linear congruences, the chinese remainder theorem, polynomial congruences, fermat's theorem, euler's totient function, and euler's theorem.

Typology: Study notes

Pre 2010

Uploaded on 03/28/2010

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Math 104A, Number Theory, Fall 2002.
List of results we have proved during the course.
1. Induction. Pro of of formulas and inequalities like 1 + 2 + · ··+n=n(n+ 1)/6 and
12+ 22+· ·· +n2=n(n+ 1)(2n+ 1)/6 and n2< n! for n > 3.
2. The binomial coefficients. (a). Defining µn
r=n!
r!(n−r)!, then
(x+y)n=µn
0xny0+µn
1xn−1y+. . . +µn
nx0yn.
(b). The binomial coefficients are integers.
(c). If pis prime and 1 ≤r≤p−1 then p¯
¯µp
r.
3. Division. Basic properties of divisibility and the division theorem.
4. The floor function (a). Definition of the floor function, basic properties like
bx+yc ≥ bxc+byc.
(b). If pis prime, the highest power of pwhich divides n! is
¹n
pº+¹n
p2º+. . .
5. There are infinitely many primes of the form 4k+ 3, 3k+ 2, 2rk+ 1 for rfixed,
2pk + 1 for a fixed prime p.
6. Greatest common divisor. (a). There exist xand ysuch that (a, b) = xa +by.
(b). If a|bc and (a, b) = 1 then a|c.
7. Fundamental Theorem of Arithmetic. Every number greater than 1 is either
prime or a product of primes, and the list of primes in the factorization is unique up
to reordering.
8. Consequences of the FTA. (a). Formulas for (a, b) and [a, b] in terms of the
prime factorizations of aand bleading to results like [a, b](a, b) = |ab|.
(b). If nis not a perfect square then √nis irrational.
9. Linear diophantine equations. (a). The equation ax +by =chas a solution
(x, y) if and only if (a, b)|c.
(b). If (x0, y0) is a solution to ax +by =cthen the general solution is given by
(x, y) = (x0+kb/(a, b), y0−ka/(a, b)).
10. Basic properties of congruences. (a). If a≡a0(mo d m) and b≡b0(mod m)
the a+b≡a0+b0(mod m) and ab ≡a0b0(mod m) and ak≡(a0)k(mod m).
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Math 104A, Number Theory, Fall 2002. List of results we have proved during the course.

  1. Induction. Proof of formulas and inequalities like 1 + 2 + · · · + n = n(n + 1)/6 and 12 + 2^2 + · · · + n^2 = n(n + 1)(2n + 1)/6 and n^2 < n! for n > 3.
  2. The binomial coefficients. (a). Defining

n r

n! r!(n − r)!

, then

(x + y)n^ =

n 0

xny^0 +

n 1

xn−^1 y +... +

n n

x^0 yn.

(b). The binomial coefficients are integers.

(c). If p is prime and 1 ≤ r ≤ p − 1 then p

p r

  1. Division. Basic properties of divisibility and the division theorem.
  2. The floor function (a). Definition of the floor function, basic properties like bx + yc ≥ bxc + byc. (b). If p is prime, the highest power of p which divides n! is

⌊ n p

n p^2

  1. There are infinitely many primes of the form 4k + 3, 3k + 2, 2r^ k + 1 for r fixed, 2 pk + 1 for a fixed prime p.
  2. Greatest common divisor. (a). There exist x and y such that (a, b) = xa + by. (b). If a|bc and (a, b) = 1 then a|c.
  3. Fundamental Theorem of Arithmetic. Every number greater than 1 is either prime or a product of primes, and the list of primes in the factorization is unique up to reordering.
  4. Consequences of the FTA. (a). Formulas for (a, b) and [a, b] in terms of the prime factorizations of a and b leading to results like [a, b](a, b) = |ab|. (b). If n is not a perfect square then

n is irrational.

  1. Linear diophantine equations. (a). The equation ax + by = c has a solution (x, y) if and only if (a, b)|c. (b). If (x 0 , y 0 ) is a solution to ax + by = c then the general solution is given by (x, y) = (x 0 + kb/(a, b) , y 0 − ka/(a, b)).
  2. Basic properties of congruences. (a). If a ≡ a′^ (mod m) and b ≡ b′^ (mod m) the a + b ≡ a′^ + b′^ (mod m) and ab ≡ a′b′^ (mod m) and ak^ ≡ (a′)k^ (mod m). 1

(b). Every integer is congruent modulo m to precisely one element of the standard residue system { 0 , 1 ,... , m − 1 }. (This is equivalent to the division theorem.)

  1. Basic applications of congruences. You can use congruences to answer several questions from Chapter 2 for which you previously used the division theorem. E.g. p. #8, p.15 #27, p.37 #8, #9.
  2. Inverses Modulo m. The number x has an inverse modulo m if and only if (x, m) = 1. The inverse of x is unique modulo m.
  3. Linear Congruences. You can solve ax ≡ b (mod m) if and only if (a, m)|b. Set d = (a, m) then if d|b, the solutions of ax ≡ b (mod m) are the solutions of (a/d)x ≡ (b/d) (mod m/d). Modulo m there are precisely d = (a, m) solutions and if x 0 is one

solution then all the solutions modulo m are x 0 , x 0 + md , x 0 + (^2) dm ,... , x 0 + (d−d1) m.

  1. Chinese Remainder Theorem.
  2. Generalized Chinese Remainder Theorem. Theorem 3.3.4.
  3. Polynomial Congruences for prime power modulii. Theorem 3.4.6 (If you know the roots of f (x) ≡ 0 (mod pk) then you can use these to find the roots of f (x) ≡ 0 (mod pk+1).)
  4. Fermat’s Theorem.
  5. Application of Fermat’s theorem. Proposition 4.1.5: If p is prime and xr^ ≡ 1 (mod p) then x(r,p−1)^ ≡ 1 (mod p).
  6. Euler’s Totient Function. If p is prime then φ(pk) = pk(1 − 1 /p). If (m, n) = 1 then φ(mn) = φ(m)φ(n).
  7. Euler’s Theorem. The proof uses the following: If x 1 ,... , xφ(r) are the numbers between 0 and m which are relatively prime to m and (a, m) = 1, and yj is the remainder when axj is divided by m, then the list of numbers y 1 ,... , yφ(m) is just a re-ordering of the list x 1 ,... , xφ(m).
  8. Lagrange’s Theorem. We proved an extended version: If f (x) is a polynomial and p is prime then f (x) = (x − r 1 ) · · · (x − rk)g(x) (poly mod p) where g has no roots modulo p. The roots of f are precisely r 1 ,... , rk. (The roots may be repeated more than once in this list.) By comparing degrees modulo p, the number of roots modulo p is at most the degree of f.
  9. Counting Roots. If p is prime and d > 0, the equation xd^ ≡ 1 (mod p) has precisely (d, p − 1) solutions modulo p. The equation xd^ ≡ −1 (mod p) has precisely (2d, p − 1) − (d, p − 1) solutions modulo p.
  10. RSA. If n = pq where p and q are distinct odd primes, and if de ≡ 1 (mod φ(n)) then mde^ ≡ m (mod n) for 0 ≤ m < n.

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