9.3. Proof of Bezout’s Theorem

Bezout’s theorem states that the GCD of two numbers can be expressed as a linear combination of those two numbers. Specifically, given two integers a and b, there exist integers s and t such that GCD(a, b) = sa + tb.

Bezout's theorem tells us an important relationship between two integers (let's call them a and b). It claims that if you can find the GCD (greatest common divisor) of these two numbers, you can always express this GCD as a combination of a and b, using integer multipliers s and t. This means you can write the GCD as some multiple of a added to some multiple of b. For instance, if a = 6 and b = 14, the GCD is 2, and you can represent 2 as a linear combination of 6 and 14 using the integers -2 and 1 (i.e., 2 = -26 + 114). This theorem is valuable in number theory and has applications in algebra and cryptography.

Imagine two people trying to share a certain number of apples and oranges (representing a and b). The GCD represents the largest number of complete fruit baskets they can create with the apples and oranges they each possess. According to Bezout’s theorem, they can combine their fruits in specific proportions (s and t) to create that maximum number of baskets.

To prove Bezout's theorem, we will define a set S that contains all integer linear combinations of a and b. This set S is infinite due to the arbitrary integers involved.

The proof of Bezout's theorem begins with defining a set S, which includes every possible integer combination of a and b. This means S consists of expressions of the form 'xa + yb', where x and y can be any integers, including negatives. Because there is no limit to how large or small x and y can be, the set S is infinite. The goal is to demonstrate that the GCD of a and b (let's call it d) is included in this set S.

Think of set S as a recipe book where x and y represent different ingredients in any quantity you desire. Since you can add as many or as few of any ingredient as you want, the possibilities for recipes (or combinations of a and b) are endless.

Claim 1 states that the set S contains non-zero elements. By choosing specific integer values for the linear combiners, we can show that both a and b belong to S, confirming the existence of non-zero elements.

The first claim establishes that the set S is not only about the zero combination (where both x and y are zero). By setting x = 1 and y = 0, you find that '1a + 0b' equals a, which is clearly a non-zero element of S (unless a is zero). Similarly, setting x = 0 and y = 1 yields b. Therefore, both a and b are in S, confirming that there are non-zero elements present.

Imagine you are collecting items. If you can choose to collect just apples (a) or just oranges (b), this shows that you have items (non-zero elements) in your collection, reinforcing that your collection is richer than just the emptiness of not collecting anything at all.

Claim 2 asserts that the element s (the one with the least absolute value in S) divides every element of S, including any arbitrary element u from S.

In this claim, s represents the smallest non-zero element in S. The claim is powerful because if s divides u (any element of the set S), it means for every element u, you can express it in terms of s and some integer quotient. This involvement of s will later help prove that d must be a divisor of all elements in S, including the GCD.

Think of s as your smallest coin denomination, like a penny. If you can buy anything with this penny (any element of S), then logically, any amount you spend is essentially divisible by that penny, ensuring you always have something tangible to trade — reinforcing the concept of divisibility.

Claim 3 states that the value s is a divisor of the GCD of a and b. Given that s divides both a and b, it follows that s must also be a divisor of their GCD.

Since we established in claim 2 that s divides every element in S, and s divides both a and b (elements of S), it follows that s is a common divisor of a and b. This leads us to conclude that s must also divide the GCD (as it is the largest common divisor). Therefore, this claim establishes a direct link between s and the GCD.

Consider a situation where a group of friends all contribute to buy a present (GCD). If the smallest contributing friend (s) has contributed to the present, it makes sense that their contribution must be a part of the total purchasing power from all friends to buy the present.

In Claim 4, we demonstrate that the GCD d divides every integer linear combination of a and b, including the specific combination that gives us s.

Claim 4 states that since d is a common divisor of a and b, it divides any linear combination of a and b. Therefore, d must divide s because s is itself a linear combination of a and b. This is significant, as it allows us to conclude that d and s are either equal or negatives of each other, leading us to express the GCD as a linear combination of a and b, confirming Bezout's theorem.

Imagine the GCD as a large pot that can hold soup made from two ingredients (a and b). If other smaller pots (s) can also fit into this large pot, it's logical to conclude that the larger pot can accommodate all combinations of soup made from those two ingredients, thus confirming that any ratio of ingredients will yield a soup that fits in the main pot.

By establishing the existence of integers s and t such that GCD(a, b) is expressible as sa + tb, we conclude that Bezout’s theorem is proven.

The final conclusion reached after the four claims is that through logical steps, we've shown the existence of integers s and t such that the GCD of a and b can be expressed as a linear combination of a and b. Although we haven't constructed specific s and t values, we confidently state their existence as guaranteed by the theorem.

This conclusion is similar to finding a recipe that ensures you can make the perfect dish with any two ingredients. Just because we haven't measured exact amounts yet doesn't mean the recipe (the linear combination) doesn't exist. We can always find those specific amounts (s and t) later through practical trial and error, just like applying the theorem in real-world scenarios.

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