Suppose that a and b are positive integers such that a 2 </msup> = 3 b

Mohammad Cannon 2022-06-28 Answered
Suppose that a and b are positive integers such that a 2 = 3 b 2 . Show that 0 < b < a < 2 b and ( 3 b a ) 2 = 3 ( a b ) 2
Use these and the Well-Ordering Principle to prove that no such a and b exist. From this it follows that 3 Q .
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Answers (2)

scoseBexgofvc
Answered 2022-06-29 Author has 20 answers
Step 1
Typical trick due to Hardy:
(assuming in lowest terms) 3 = a b 3 3 = 3 b a (Componendo et Dividendo) 3 = 3 b a a b
Since 1 < 3 < 2, b < a < 2 b 0 < a b < b
Now 3 b a a b is lower than a b , contradiction.

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seupeljewj
Answered 2022-06-30 Author has 7 answers
Step 1
Fairly common to demand the a 2 = 3 b 2 be the solution in positive integers that has the smallest possible value of a + b
Step 2
But then they point out ( 3 b a ) 2 = 3 ( a b ) 2 .. You are required to show that a b , 3 b a are both positive integers, and then show that 3 b a + a b < a + b ,, which is a contradiction of the assumption that a + b was minimal

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