What is the largest square pyramidal number that is also a prime number?
Clarification:
Where
is the
-th square pyramidal number.
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We can easily see that P 2 = 5 is prime. Now we need to prove that there are no larger square pyramidal numbers which are prime.
The closed form expression for square pyramidal numbers is P n = 6 n ( n + 1 ) ( 2 n + 1 ) .
P 3 = 1 4 is not a prime. Now if n > 3 , n + 1 > 4 and 2 n + 1 > 7 .
Now if we divide n ( n + 1 ) ( 2 n + 1 ) by 6 we are left with a product of at least 2 integers which are both greater than 1 because each of n , n + 1 , 2 n + 1 are greater than any of 2 , 3 .
Since a product of two integers greater than 1 cannot be a prime. There cannot be a larger square pyramidal number which is prime.
Hence the answer is 5 .