Math 307 – Number Theory Study Guide
Theorem 1.2: Axiom of Well-Ordering (1st Principle of Finite Induction)
Every non-empty subset 'S' of a set of (non-negative) integers S will contain a LEAST
element; that is there is an element 'a' in 'S' such that 'a ≤ b' for every 'b' in 'S'
Theorem 1.3: The Second Principle of Finite Induction
Let 'S' denote a subset of the positive integers with the following properties:
(a) 1 is an element of 'S'
(b) If 1, 2, 3,..., 'k-1' are in 'S', then 'k' is an element of 'S'
Then 'S' are all positive integers
The Splitting Identity
[1/k] = [1/(k+1)] + [1/k(k+1)]
Binomial Theorem
nCk = (nk) =
(a+b)n
=
[ n!/k!(n-k)! ]
n
∑( k)an-k ∙ bk
= 2n
Pascal’s Rule:
(nk) + (nk-1) = (n+1k) for 1 ≤ k ≤ n
Chapter 2: Divisibility Theory in the Integers
Theorem 2.1: Division Algorithm
If ‘a’ & ‘b’ are integers where b > 0, then there exist unique integers ‘q’ & ‘r’ such that:
a = q∙b + r 0 ≤ r < b
Definition 2.1:
‘b’ is divisible by ‘a’ (a ≠ 0) provided there is some integer ‘c’ such that
b = a ∙ c  (a | b)
Theorem 2.2:
(a) (a | 0), (1 | a), (a | a)
(b) (a | 1) iff a = +/-1
(c) If (a | b) and (c | d), then (ac | bd)
(d) If (a | b) & (b | c), then (a | c)
(e) (a | b) and (b | a) iff a = +/- b
(f) If (a | b) and b≠0, then |a| ≤ |b|
(g) If (a | b) and (a | c), then (a | (bx + cy)) for arbitrary integers ‘x’ & ‘y’
Definition 2.2:
‘d’ is the Greatest Common divisor of ‘a’ & ‘b’ provided:
(1.) (d | a) & (d | b)
(2.) If (c | a) & (c | b), then c ≤ d
Theorem 2.3:
Given integers ‘a’ & ‘b’, not both of which are zero, there exist integers ‘x’ & ‘y’ such that ‘
gcd(a,b) = a∙x + b∙y
Theorem 2.5: Euclid’s Lemma
If (a | bc), with gcd(a,b) = 1, then (a | c).
Euclidean Algorithm:
18x + 5y = 48 
18 = 3(5) + 3
5 =3+2
3 = 2 + 1  gcd(18,5) = 1
2 = 2(1)
Theorem 2.9: Diophantine Equation
The equation (a∙x + b∙y = c) has a solution iff (gcd(a,b) | c). If x0 , y0 is any particular
solution of this equation, then all other solutions are given by:
x = x0 + (b/d)∙t
y = y0 + (a/d)∙t where t is an arbitrary integer
Chapter 3: Primes and their Distribution
Theorem 3.1:
Let ‘p’ be a prime. If (p | ab), then (p | a) or (p | b)
The only possible divisors of prime ‘p’, are 1 & p. Since p X a, gcd(p,a) ≠ p.
Theorem 3.2: Fundamental Theorem of Arithmetic
Every positive integer n > 1 can be written as a product of primes. This representation is
unique, apart from the order in which the factors occur.
Theorem 3.3: Pythagoras’ Theorem
The number √2 is irrational
The Sieve of Eratosthenes:
In testing the primality of a specific integer a > 1, it is suffice to divide a by those primes
not exceeding √a.
Theorem 3.4: Euclid’s Infinite Primes
There are an infinite number of primes
Theorem 3.5:
If pn is the nth prime number, then pn ≤ 22n-1
Chapter 4: Theory of Congruences
Definition 4.1:
Let ‘n’ be a fixed positive integer. Two integers are said to be congruent modulo n:
a ≡ b (mod n)
Theorem 4.1:
For arbitrary integers a and b, a ≡ b (mod n) iff ‘a’ and ‘b’ leave the same nonnegative remainder
when divided by n.
Theorem 4.2:
Let n > 1 be fixed and a,b,c,d be arbitrary integers. Then the following properties hold:
(a)
a ≡ a (mod n)
(b)
If a ≡ b (mod n), then b ≡ a (mod n)
(c)
If a ≡ b (mod n) and b ≡ c (mod n), then a ≡ c (mod n).
(d)
If a ≡ b (mod n) and c ≡ d (mod n), then a+c ≡ b+d (mod n) && ac ≡ bd (mod n)
(e)
If a ≡ b (mod n), then a+c ≡ b+c (mod n) and ac ≡ bc (mod n)
(f)
If a ≡ b (mod n), then ak ≡ bk (mod n) for any int k > 0
Theorem 4.3:
If ca ≡ cb (mod n), then a ≡ b (mod n/d), where d = gcd(c,n)
Corollary 1:
If ca ≡ cb (mod n) & gcd(c,n)=1, then a ≡ b (mod n)
Corollary 2:
If ca ≡ cb (mod p) and p |/ c, where p is a prime number, then a ≡ b (mod p)
Theorem 4.7:
The linear congruence ax ≡ b (mod n) has a solution iff (d | b), where d = gcd(a,n). If
(d | b), then it has ‘d’ mutually incongruent solutions modulo ‘n’.
Corollary:
If gcd(a,n) = 1, then the linear congruence ax ≡ b (mod n) has a unique solution mod ‘n’.
Skip 4.4
Chapter 5: Fermat’s Theorem
Fermat Factorization:
10541  1022 = 10404 < n = 10541 < 1032
1032 – n = 10609 – 10541 = 68  NOT a square
1052 – n = 11025 – 10541 = 484 = 222
Therefore: (105 - 22)(105 + 22) = 83∙22 = n = 1052 – 222
Theorem 5.1: Fermat’s Theorem
Let ‘p’ be a prime and suppose that p |/ a. Then ap-1 ≡ 1 (mod p)
Corollary:
If ‘p’ is a prime, then ap ≡ a (mod p) for any integer a.
Lemma:
If ‘p’ and ‘q’ are distinct primes with ap ≡ a (mod q) and aq ≡ a (mod p),
then apq ≡ a (mod pq)
Theorem 5.2:
If ‘n’ is an odd pseudoprime, then Mn = 2n – 1 is a larger one.
Theorem 5.3:
Let ‘n’ be a composite square-free integer, say n = p1p2…pr, where the pi are distinct primes. If
[(pi -1) | (n-1)], for i = 1,2,…,r, then ‘n’ is an absolute pseudoprime.
Theorem 5.4: Wilson’s Theorem
If ‘p’ is a prime, then (p-1)! ≡ -1 (mod p)
Theorem 5.5:
The quadratic congruence x2 + 1 ≡ 0 (mod p), where ‘p’ is an odd prime,
has a solution iff p ≡ 1 (mod 4)
Chapter 7: Euler’s Generalization of Fermat’s Theorem
Definition 7.1:
For n ≥ 1, let Φ(n) denote the number of positive integers not exceeding ‘n’ that are relatively
prime to ‘n’
Theorem 7.1:
If ‘p’ is a prime, and k > 0, then:
Φ(pk) = pk – pk-1 = pk(1 – 1/p)
Lemma:
Given integers, ‘a,b,c’ gcd(a,bc) = 1, iff gcd(a,b) = 1 & gcd(a,c) = 1.
Theorem 7.2:
The function Φ is a multiplicative function
Theorem 7.3:
If the integer n > 1 has the prime factorization n = p1k1 p2k2 … prkr , then:
Φ(n) = (p1k1 - p1k1-1) (p2k2 – p2k2-1) … (prkr – prkr-1) = n(1 - 1/p1) (1 - 1/p2)… (1 - 1/pr)
Theorem 7.4:
For n > 2, Φ(n), is an even integer.
Theorem 7.5: Euler’s Theorem
If n ≥ 1 & gcd(a,n) = 1, then aΦ(n) ≡ 1 (mod n)
Corollary:
If ‘p’ is a prime, and p |/ a, then ap-1 ≡ 1 (mod p)
Skip 7.4
k
RSA Encoding: M ≡ r (mod n)
RSA Decoding: kj ≡ 1 (mod Φ(n))
Chapter 10: Perfect Numbers
Definition 10.1:
A positive integer ‘n’ is said to be perfect if ‘n’ is equal to the sum of all its positive divisors,
excluding ‘n’ itself.
Theorem 10.1:
If 2k-1 is a prime,(k >1), then n = 2k-1(2k – 1) is perfect and every even perfect number is this form
Lemma:
If ak – 1 is a prime (a > 0, k ≥ 2), then a = 2 and ‘k’ is also prime.
Theorem 10.2:
An even perfect number ‘n’ ends in the digit 6 or 8; equivalently, either n ≡ 6 (mod 10)
or n ≡ 8 (mod 10). **(NOTE: n ≡ 28 (mod 100))**
Definition 10.3: Mersenne Numbers
Mn = 2n – 1 for n ≥ 1
Theorem 10.3:
If ‘p’ and q = 2p + 1 are primes, then either (q | Mp) or (q | Mp + 2), but not both.
Theorem 10.4:
If q = 2n + 1 is prime, then we have the following.
(a)
(q | Mn), provided that q ≡ 1 (mod 8) or q ≡ 7 (mod 8)
(b)
(q | Mn + 2), provided that q ≡ 3 (mod 8) or q ≡ 5 (mod 8)
Corollary:
If ‘p’ and q = 2p + 1 are both odd primes, with p ≡ 3 (mod 4), then (q | Mp)
Theorem 10.5:
If ‘p’ is an odd prime, then any prime divisor of Mp is of the form 2kp + 1.
Theorem 10.6:
If ‘p’ is an odd prime, then any prime divisor of ‘q’ of Mp is of the form q ≡ ±1 (mod 8)
Chapter 11: The Fermat Conjecture
Definition 11.1:
A Pythagorean triple is a set of three integers x, y, z such that x 2 + y2 = z2; the triple is said to be
primitive if gcd(x,y,z) = 1.
Lemma 1:
If x,y,z is a primitive Pythagorean triple, then one of the integers ‘x’ or ‘y’ is even, while
the other is odd.
Lemma 2:
If ab = cn, where gcd(a,b) = 1, then ‘a’ and ‘b’ are nth powers; that is, there exist positive
integers a1,b1, for which a = a1n, b = b1n.
Theorem 11.1:
All the solutions of the Pythagorean equation:
satisfying the conditions:
are given by the formulas:
for x > t > 0, such that gcd(s,t) = 1, & s
x2 + y2 = z2
gcd(x,y,z)=1, 2|x, x>0, y>0, z>0
x = 2st, y = s2 – t2, z = s2 + t2
≠ t (mod 2)
Theorem 11.2:
The radius of the inscribed circle of a Pythagorean triangle is always an integer.
Theorem 11.3:
The Diophantine equation x4 + y4 = z2 has no solution in positive integers x,y,z.
Corollary:
The equation x4 + y4 = z4 has no solution in the positive integers.
Theorem 11.4:
The Diophantine equation x4 - y4 = z2 has no solutions in the positive integers x,y,z.
Theorem 11.5:
The area of a Pythagorean triangle can never be equal to a perfect (integral) square.
Chapter 12: Representation of Integers as Sums of Squares
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