You Will Need a Square State Of Mind

Number Theory Level 4

What is the sum of the 2 \displaystyle 2 smallest possible values of n ∈ Z \displaystyle n \in \mathbb{Z} , such that both 2 n + 1 \displaystyle 2n+1 and 3 n + 1 \displaystyle 3n+1 are perfect squares?

Bonus Find the GCD of all such n n .


The answer is 40.

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Satvik Golechha by Satvik Golechha , 21, India

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3 solutions

Vinod Kumar
Feb 11, 2015

2 n + 1 2n+1 is a square so it is odd square.

Now list of odd squares are 1 , 9 , 25 , 49 , 81 , … 1,9,25,49,81, \dots , from this list both 1 ( 1 = 2 × 0 + 1 , 3 × 0 + 1 = 1 2 ) , 81 ( 81 = 2 × 40 + 1 ) , ( 3 × 40 + 1 = 121 = 1 1 2 ) (1=2\times 0+1, 3\times 0+1=1^{ 2 }),81(81=2\times 40+1),(3\times 40+1=121=11^{ 2 } ) satisfy the condition so sum is 0 + 40 = 40 0+40=40 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

2 n + 1 2n+1 is a square so it is odd square.

Now list of odd squares are 1 , 9 , 25 , 49 , 81 , … 1,9,25,49,81, \dots , from this list both 1 ( 1 = 2 × 0 + 1 , 3 × 0 + 1 = 1 2 ) , 81 ( 81 = 2 × 40 + 1 ) , ( 3 × 40 + 1 = 121 = 1 1 2 ) (1=2\times 0+1, 3\times 0+1=1^{ 2 }),81(81=2\times 40+1),(3\times 40+1=121=11^{ 2 } ) satisfy the condition so sum is 0 + 40 = 40 0+40=40 .

Can you copy the LaTeX \LaTeX off my comment , it'll look better this way ? Also, I didn't mean to be rude if you feel that way , I just wanted your solution to look good . ⌣ ¨ \ddot\smile . Please check if I made any error and tell me .

A Former Brilliant Member - 6 years, 3 months ago

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Thanks! Done!

Pranjal Jain - 6 years, 2 months ago

Yes, what about the other values of n and the gcd of all such values?

Abhisek Mohanty - 6 years, 2 months ago

what if we put 4 and 1

Avi c - 6 years, 3 months ago
Jehad Aly
Mar 2, 2015

Can anyone please tell me the gcd of n? Gcd(0,40) is 40, but what about the other values of n?

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Brock Brown
Feb 16, 2015

Python: 

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def perfect(n):
    a = 0
    while a**2 <= n:
        if a**2 == n:
            return True
        a += 1
    return False
def goal(n):
    if perfect(2*n+1):
        if perfect(3*n+1):
            return True
    return False
n = 0
count = 0
total = 0
while count < 2:
    if goal(n):
        count += 1
        total += n
    n += 1
print "Answer:", total

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice Python script, I think that this equation is also quite well made

Marcelo Carpenter - 4 years, 2 months ago

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