设正整数 a, b 满足：
a^2 + b^2 \mid a^3 + b^3
求证：
\frac{a^3 + b^3}{a^2 + b^2} = a + b
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We are given positive integers a and b such that a squared plus b squared divides a cubed plus b cubed. Our goal is to prove that the quotient of a cubed plus b cubed divided by a squared plus b squared equals a plus b. This is a classic number theory problem that combines divisibility theory with algebraic manipulation techniques.
To solve this problem, we need to use the sum of cubes factorization formula. We know that a cubed plus b cubed equals the product of a plus b and a squared minus ab plus b squared. Applying this to our expression, we get the quotient equals a plus b times the fraction a squared minus ab plus b squared over a squared plus b squared. This transformation is crucial for our proof.
Now we substitute our factorization into the original expression. We have the quotient equals a plus b times the fraction a squared minus ab plus b squared over a squared plus b squared. For this entire expression to equal a plus b, the fraction must equal 1. This means a squared plus b squared must divide a squared minus ab plus b squared. This is our key equivalence that we need to prove.
Now we develop the key insight by examining the relationship between a squared plus b squared and a squared minus ab plus b squared. Let's calculate their difference. We get a squared plus b squared minus a squared plus ab minus b squared, which simplifies to just ab. This means if a squared plus b squared divides a squared minus ab plus b squared, then it must also divide their difference, which is ab.
We conclude by analyzing when a squared plus b squared can divide ab. For this divisibility to hold, we need a squared plus b squared to be less than or equal to ab. However, for positive integers, a squared plus b squared is always greater than ab. This contradiction means that whenever the original divisibility condition holds, we must have a squared plus b squared equals a squared minus ab plus b squared, which gives us our desired result that the quotient equals a plus b.