Is this true? There is no prime number p congruent to 1 modulo 4 which is also divisible by 3. Any proofs?

Question

A prime number is an integer p whose only positive divisors are 1 and p itself.

I made an error with phrasing the question. The question should have been: Is this true? There is no prime power p^k with p congruent to k and congruent to 1 modulo 4, which is also divisible by 3. Any proofs?

Answer 1

I deal with the truth all the time. This seems like one of those things i dont deal with.

Answer 2

If a prime power p^k is divisible by three then p = 3. The odd power of 3 are congruent to 3 mod 4, the even powers of 3 are congruent to 1 mod 4.

Answer 3

9 isn't a prime number, but it's congruent to 1 modulo 4 and it's divisible by 3. But by definition, apart from 3 itself, no prime number is divisible by 3.

Answer 4

As you said, any prime number has exactly two divisors (1 and itself). If this number is divisible by 3, then it must necessarily be equal to 3, which is NOT congruent to 1 modulo 4. So, in fact, there is no such primer number.

Answer 5

p congruent to 1 modulo 4 and p congruent to 1 modulo 3 is 13.

p congruent to k modulo 4, k=1 is one answer 5 is another, 9 is another.
Are you sure of your question?

Answer 6

erm...only 1 prime is divisible by 3 and that is 3 itself.
3 not congruent to 1 modulo 4, therfore we are done (QED)

Answer 7

Since the numer would have to be divisible by 3 and 4, yes, that is true. There are no prime numbers that can fit that description.

Answer 8

By definition p is never divisible by 3.
Except p=3 which is 3 =3 mod4 and this is not 3 = 1 mod 4.
So the proof was given.
Th

Answer 9

It is still unclear, what should be divisible by 3 ?
if it is p^k is 0 mod 3, then it is obvious, p=3 since p is prime, but 3 is incongruent 1 mod 4 thus p=3 is no solution. done,

Answer 10

The Manfreds. The song was even called 5-4-3-2-1, which I think was very nice of them.