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7 solutions
Nice solution!
P = 3^2 - 4 = 5 .... don't worry by fix it
What about 29 is a prime and 29-4=25 which is perfect square so you are wrong.
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What did u mean to say?
You don't understand.
P.S. it is 4 LESS than a square, not four MORE :)
Why -3 is not considered a solution?
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negative number can't be a prime number
1 isn't a prime number because a prime number needs two integer factors.
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Yes .... 1 y P. P = (n + 2)(n - 2) (n + 2) = 1 or (n - 2) = 1 That's what he said and is correct.
I think -3 is a prime Number, so -3 + 4= 1, and 1 is a square
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@Waldemar Barrera – Yeah but -3 is divisible by 1, 3, -1 and -3, there fore making it a composite number.
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@Herbert Stewart – Is the same for every prime Number p, the divisors aré 1,-1, p,-p. I think the trouble is define the prime Numbers only positives Numbers.
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@Waldemar Barrera – Lemme make it clear :)
"A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself."
But all negative numbers have divisor as − 1 as well as they are less than 1 , thus , none of negative number is a prime :)
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@Nihar Mahajan – Ahhhh. That actually makes it very clear for me. Thanks Nihar!
@Nihar Mahajan – Nihar,I think if you're considering negative factors then − 1 , − p are also factors of a prime p .So thought of in that way,each prime number has 4 factors.I think a prime number is simply defined to be a natural number.
- 3 is not a prime no. because a prime no. defines as a natural no greater than 1.When we say natural no. it means commonly in nature and an obvious number therefore natural.no. is referring to whole no. And non-negative.no.
what about 221+4=225
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2 2 1 is not a prime my friend...
13*17 = 221
I thought this once in my mind( I know this is not a valid solution and I shouldn't be posting it anyways) and was amazed to find that only one such number existed, increased my respect for 5...
Similar to @Nihar Mahajan , but with more detailed explanation:
Let the prime number be p . Then p + 4 = n 2 which means that p = n 2 − 4 = ( n − 2 ) ( n + 2 ) . Considering that p is a prime, should ( n − 2 ) be greater than 1, then p would have two non-trivial factors (factors other than 1 and p itself) which violates the definition of a prime. Thus n − 2 = 1 which means n = 2 + 1 = 3 and in turn n + 2 = 3 + 2 = 5 . Finally p = 5 and indeed this is the ONE and only one such prime.
5+4=9 which is a perfect square
5 + 4 = 9 is the only solution I can find which fits. Let some prime p plus 4 equal a perfect square, so p + 4 = x squared. Therefore p = x squared -4. Now since this is equal to a prime, one of the factors must be x and the other must be 1. For the case of 1, 1 + 4 squared is 25, which is trivially a perfect square.
Good .... I like prime number. What is the significance or meaning for you? (P - 1)!/P! .... Where P is prime number and ( ! ) is prime factorial. (2 - 1)/2 * (3 - 1)/3 * (5 - 1)/5 * .....and so
Let the prime number be p . Then we have p + 4 = n 2 for some integer n .
p + 4 = n 2 ⇒ p = n 2 − 4 ⇒ p = ( n + 2 ) ( n − 2 )
Since p is a prime , its smallest factor is 1 which means n − 2 = 1 ⇒ n = 3 ⇒ p = 3 2 − 4 = 9 − 4 = 5 .
There is also a possibility that p = ( − 1 ) × ( − p ) and in this case since − 1 > − p , n + 2 = − 1 ⇒ n = 3 ⇒ p = 3 2 − 4 = 9 − 4 = 5 .
Thus in both cases , p = 5 is only solution and the total number of solutions possible is 1 .