Just trial and error would be fine.

Number Theory Level 1

Find the second smallest prime p p such that p − 1 2 \sqrt{\frac{p-1}{2}} is an integer.

Tips: the smallest such prime is 3 3

Hoping accuracy 100 % 100\% !


The answer is 19.

Culver Kwan by Culver Kwan , 16, Hong Kong

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2 solutions

Culver Kwan
Mar 19, 2020

p − 1 2 \frac{p-1}{2} is a square number.

If p − 1 2 = 1 \frac{p-1}{2}=1 , p = 3 p=3 which is our smallest solution.

If p − 1 2 = 4 \frac{p-1}{2}=4 , p = 9 p=9 which is not a prime.

If p − 1 2 = 9 \frac{p-1}{2}=9 , p = 19 p=19 which is our second smallest solution.

So the answer is 19 \boxed{19} .

Easy?

2 Helpful 0 Interesting 0 Brilliant 0 Confused

Hlo.what are the squares that you remember 1,4,9,16...... Double them 2,8,18,32..... Add one 3,9,19,33... Which are prime Hmmm 3,19..... So,here is your solution Cheers!

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