Alvin’s Theorem

While Ms. Powers was leading a class discussion about square numbers, Absent-minded Alvin was in another “world”, looking for interesting patterns in the topic. Shortly, he raised his hand and said, “Ms. Powers, I’ve found something rather nice. Look. If I take 2 consecutive squares and subtract them, the difference is always the sum of 2 consecutive integers.””Show the class what you mean by that, Alvin,” said the teacher.

Alvin wrote the following on the board:

49 – 36 = 13 and 13 = 6 + 764 – 49 = 15 and 15 = 7 + 8

Turning to the class, he shyly said, “I call this ‘Alvin’s Theorem‘.”

Ms. Powers smiled and said, “Very good, but if you want to call it a theorem, you must be able to prove it is always true for all numbers, using algebra.”

Alvin replied, “Oh yes, I can do that too. Here’s how.”

What did Alvin write on the board now?

Later, Alvin investigated the matter of the difference of consecutive “even” squares. What do you imagine he discovered this time?

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