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By Alexander Soifer

This moment variation of Alexander Soifer’s How Does One lower a Triangle? demonstrates how diversified components of arithmetic could be juxtaposed within the answer of a given challenge. the writer employs geometry, algebra, trigonometry, linear algebra, and earrings to increase a miniature version of mathematical examine.  

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Let us take a closer look at the equation px + qy = 0. 3 It certainly has a solution at x = 0, y = 0. In addition, the condition (∗) above requires this equation to have at least one more solution in the integers, namely x = m, y = −n. Such situations appear quite often in mathematics. We have a special definition for them: If the equation px + qy = 0 (1) has a non-zero solution in integers, then the numbers p, q are called integrally dependent. If equation (1) above has only the zero solution x = y = 0, then the numbers p, q are called integrally independent.

6(2xy) + 1(2x2 + 3y2 − 4z2 ) = 0. 5). Otherwise, prove it now. 1, the numbers 6 and 1 are integrally independent; thus, the equality (5) implies 2xy = 0 and 2x2 + 3y2 − 4z2 = 0. The first one requires x = 0 or y = 0, in contradiction with our assumption. We proved that x = y = z = 0 is the only integral solution to √ √ √ equation (3); therefore, 2, 3, 4 are integrally independent. 2. Prove that three angles of a right triangle are integrally dependent. Solution. , α(1) + β(1) + γ(−1) = 0. The last equality shows that the equation αx + βy + γz = 0 has a non-zero solution in integers, and thus completes the proof.

We have a special definition for them: If the equation px + qy = 0 (1) has a non-zero solution in integers, then the numbers p, q are called integrally dependent. If equation (1) above has only the zero solution x = y = 0, then the numbers p, q are called integrally independent. In Pool Table Problem I, the ball will get into a pocket of a p × 2q pool table if and only if the system of numbers p, q is integrally dependent! Try your pool table skills on the next problem: Pool Table Problem II. In the setting of Pool Table Problem I, let p and q be odd integers.

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