xouper wrote: ↑Thu Dec 27, 2018 2:18 amThat springs forth a multitude of thoughts:
- They start out saying "Any odd prime" and then later say, wait, no, it only applies to certain odd primes.
- There are only two kinds of odd prime numbers, those that are "congruent to 1 modulo 4", and those that are "congruent to 3 modulo 4".
- Abdul will appreciate the use of the word "modulo" and ed will be confused by it.
- There are no even prime numbers that are "congruent to 1 modulo 4", therefore the theorem is true for any prime number, not just odd prime numbers.
- Mathematicians sometimes have a funny way of expressing theorems.
- For the non-mathematician, the phrase "congruent to 1 modulo 4" means if you divide a number by 4 the remainder is 1. Example: 13 divided by 4 has a remainder of 1.
- For any whole number, if you divide it by 4, the remainder will be either 0, 1, 2, or 3. Those are the only possibilities.
- There is only one prime number that is "congruent to 2 modulo 4", and that is 2. That is to say, 2 divided by 4 has a remainder of 2.
- There are no prime numbers that are "congruent to 0 modulo 4". That is to say, there are no prime numbers that are divisible by 4 with no remainder.
- The phrase "congruent to 1 modulo 4" also implies there exists a number x such that p = 4x + 1. This should be obvious because if you divide 4 into (4x +1) then you get a remainder of 1.
- Shemp has absolutely no choice but to live the rest of his life by those words, irregardless what he might want to do instead.
- It is not necessary to square the p. In fact it is redundant to do so, since if p is equal to the sum of two squares, then so is p2.
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