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AT Math Unit 1 Definitions and Statements

Terms in this set (17)

Bezout's Identity
For any two nonzero integers a and b, there exists a pair of integers x and y solving for the equation ax + by = (a,b), where (a,b) is the greatest common divisor of a and b
Diophantine Equation
An equation involving polynomials in two or more variables with integer coefficients. In general, we only seek solutions for which all variables take integer values
Primitive Pythagorean Triple
Triples of the form x²+y²=z² where the greatest common divisor is 1
The Well-Ordering Property of N
All non-empty sets of positive integers have a least element
The Two Out of Three Principle for Divisibility
Condition: a + b = c. If any two numbers from the set {a, b, c} are divisible by d, then the third number must be divisible by d
Prime Number
A positive integer p such that p|ab for some product of integers ab, then it must necessarilt be the case that p|a or p|b
Composite Number
A natural number which is not prime or equal to 1
Relatively Prime
A pair or tuple of integers is relatively prime if their greatest common divisor is 1
Greatest Integer Function
The function assigning a real number x to the greatest integer ⌊x⌋ which is less than or equal to x
The Division Algorithm
Let a, b ∈ ℤ with b > 0. Then there exist unique q, r ∈ ℤ such that a = bq + r, 0 ≤ r < b
The Greatest Common Divisor
The GCD of two positive integers a and b, denoted (a, b), is the largest integer d such that d | a and d | b
Euclid's Lemma
If a, b, c ∈ ℤ with (a, b) = 1 and a | bc then a | c
Pairwise Relatively Prime
Let a₁, ..., aₙ be integers, not all zero. If (aᵢ, aⱼ) = 1 for all pair where i ≠ j, 1 ≤ i, j ≤ n, then a₁, ..., aₙ are pairwise relatively prime
Pascal's Identity
Binomial Theorem
The Necessary and Sufficient Conditions to Solve the Diophantine Equation ax + by = c where a and b are both nonzero
c is a multiple of (a, b)
The General Form of the Solution to the Diophantine Equation ax + by = c, where a and b are both nonzero and when a solution exists
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