Brilliant is perfect!

Number Theory Level 2

At one point (in time X) the number of students or members in Brilliant was a perfect square. In one year 2000 more students joined and the new strength of the Brilliant website is one more than a perfect square. What was the original number of users of Brilliant during time X?


The answer is 998001.

Sathvik Acharya by Sathvik Acharya , 17, India

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1 solution

We are given that there exists positive integers m , n m,n such that

m 2 + 2000 = n 2 + 1 ⟹ n 2 − m 2 = 1999 ⟹ ( n − m ) ( n + m ) = 1999 m^{2} + 2000 = n^{2} + 1 \Longrightarrow n^{2} - m^{2} = 1999 \Longrightarrow (n - m)(n + m) = 1999 .

Now as 1999 1999 is prime and n > m ≥ 0 n \gt m \ge 0 we must have that

n − m = 1 , n + m = 1999 ⟹ ( n − m ) + ( n + m ) = 2000 ⟹ n = 1000 , m = 999 n - m = 1, n + m = 1999 \Longrightarrow (n - m) + (n + m) = 2000 \Longrightarrow n = 1000, m = 999 .

The original number of Brilliant members is then m 2 = 99 9 2 = 998001 m^{2} = 999^{2} = \boxed{998001} .

2 Helpful 0 Interesting 0 Brilliant 0 Confused

I did it the same way.

Kaushik Chandra - 3 years, 8 months ago

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