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## Can a product of 4 consecutive odds be a perfect square?

Someone on twitter asked if a product of four consecutive odd positive numbers can be a perfect square.

For example, 3 x 5 x 7 x 9 = 945 which is not a perfect square. (30^2 = 900 and 31^2 = 961.) Similarly, 7 x 9 x 11 x 13 = 9009, again is not a perfect square.

Here is my proof. Write the four consecutive positive odd numbers as:

2x+1, 2x+3, 2x+5, 2x+7

where x is a positive integer.

The middle two numbers 2x+3, 2x+5 cannot both be perfect squares since their difference is 2 — the difference between two positive consecutive perfect squares is at least 3. Let’s suppose it is 2x+3 that is not a perfect square. (The argument can still be adapted if it was 2x+5.) So 2x+3 is divisible by an odd prime p that has an odd power in its prime factorization. This prime p cannot divide the other three factors 2x+1, 2x+5, 2x+7, or else it would divide their differences from 2x+3, which are 2 or 4 (and p is odd). Therefore the prime p appears in the prime factorization of the product

(2x+1)(2x+3)(2x+5)(2x+7)

with only an odd power, hence this product cannot be a perfect square. QED

1. January 2, 2017 at 3:58 pm

-3*-1*1*3=9?

2. January 2, 2017 at 4:00 pm

Oh I missed the positive bit

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