Hello. This is Watanabe Tenmei from Seito Medical School Clear File. This time, I'm going to talk about prime numbers, and I might have found the formula for prime numbers.
The first expression that comes to mind for prime numbers is the expression a^2+b^2. By the way, a^2 means multiplying a twice, and a^3 means multiplying a three times.
I have long suspected that a^2-b^2-c^2 is a prime number formula. I couldn't say for sure, so I felt confused.
The formula a^2+b^2 yields all 1mod4 prime numbers. However, the prime number of 3mod4 does not come out. By the way, 1mod4 means that the remainder when divided by 4 is 1, and 3mod4 means that the remainder when divided by 4 is 3.
The reason why only 1mod4 comes out is that when you substitute 0, 1, 2, and 3 for a^2, 0^2=0, 1^2=1, 2^2=4, 3^3 =9. If we look at the mod4, or the remainder when divided by 4, we get 0mod4, 1mod4, 0mod4, and 1mod4.
We have looked at numbers 0, 1, 2, and 3 in order, but in the subsequent numbers 4, 5, 6, and 7, their square mod4 is 0, 1, 0, 1. If you mod4 these 4, 5, 6, 7, or any subsequent numbers, they will become 0mod4, 1mod4, 2mod4, or 3mod4, so their squares are 0mod4, 1mod4, 0mod4, and 1mod4, respectively.
Since a^2 mod4 is 0 or 1, a^2+b^2 is either (0+0), (0+1), (1+0), or (1+1). In fact, this is one of 0, 1, or 2, so it will be anything other than 3. By the way, 4 is 0mod4, so 4 and 0 are the same mod4, 0mod4 as a remainder when divided by 4.
In other words, a^2+b^2=3mod4 does not exist. At least for integers.
So, let's change the sign of a^2+b^2. a^2 – b^2 is the product of all composite numbers, that is, the product of multiplication. A prime number is a number that is only divisible by itself, such as 2, 3, 5, and 7, but a composite number can be said to be the product of two or more integers such as 5×3 or 5×7, such as 15 or 35. .
Here, since a^2 – b^2=(a+b)(ab), there are conditions, but all composite numbers will come out. The condition is that when the difference between a and b is 1, a prime number may appear, for example 4^2- 3^2 = 7. If the difference between a and b is 2 or more, and both a and b are integers that are not 0, a composite number will appear.
For example, 5^2 – 3^3 = 16, 7^2 – 2^2 = 45.
Earlier, we reversed the sign of the first equation a^2 + b^2 from + to - as a^2 – b^2, but if we change the sign, we will get the opposite value. For example, if you change 4a+1 to 4a-1, 4a+1=1,5,9,13,17,21,... and 4a-1=3,7,11,15,19,23,... Odd numbers that did not exist in 4a+1 will now appear in 4a-1.
So far, a^2+b^2 includes all prime numbers of 1mod4, and a^2-b^2 includes all composite numbers, although there are conditions. While a^2 – b^2 was a composite number, by reversing its sign, a prime number of 1mod4 appears, which is the opposite of a composite number.
So, even though we inverted the formula for the composite number by changing its sign, why doesn't the formula for the prime number 3mod4 still come out? This is because the sign of b^2 has changed between a^2 – b^2 and a^2 + b^2, but the sign of a^2 has not changed.
If we change the formula for the composite number of a^2 – b^2 to the formula for the prime number of a^2+b^2 which is 1mod4, and then change the sign of a^2, we get -a^2+b^2. , is essentially the same as the formula for a^2-b^2. However, if you change the sign of a^2 and b^2 from + to -, it becomes - a^2 – b^2, which becomes a negative value.
In other words, from the relationship of only a^2 and b^2, there is no prime number of 3mod4.
Then what should we do. Here, let's reverse the expression for a composite number, not by its sign, but by whether it is a prime number or a composite number. a^2 – b^2 Since b^2 is a composite number, let's replace b^2 with a prime number p. a^2- p Now, for this expression to be an odd prime number, a^2 must be 0mod4.
In other words, if you subtract the prime number of 1mod4 or the prime number of 3mod4 from 0mod4, 0mod4-1mod4=3mod4, 0mod4-3mod4=1mod4. By the way, 0mod4-1mod4 is 3mod4, but it is essentially the same as mod4, and when divided by 4, -1mod4=3mod4=7mod4, etc.
Here, we want to obtain a prime number of 3mod4, so by substituting the prime number formula of 1mod4, that is, b^2+c^2, to p of a^2 – p, we obtain a prime number of a^2 – p = 0mod4 – 1mod4 = 3mod4. Masu. The reason why a, b, and c are interchanged is because a^2+b^2 and b^2+c^2 are essentially the same addition of squares. As long as other people can see it and the same alphabets don't overlap, there's no problem in changing it around.
By the way, a^2+b^2 and a^2-b^2 are essentially addition and subtraction, so please be careful about that.
Now, if we substitute b^2+c^2 for p, we get a^2 – (b^2 + c^2). Expanding the formula, we get a^2 – b^2 – c^2, which is the prime number formula mentioned at the beginning. By the way, a^2 must be 0mod4 and a is an even number. b^2+c^2 is essentially the same because the combination of b and c is even and odd or odd and even. In other words, 0mod4+1mod4, so if b^2+c^2 is fixed as an even number and an odd number, a^2 – b^2 – c^2 = even number^2 – even number^2 – odd number^2.
Even numbers can be expressed as 2a and 2b, and odd numbers can be expressed as 2c+1, so it becomes (2a)^2-(2b)^2-(2c+1)^2, which is the prime number expression of 3mod4. Since the prime number expression of 1mod4 is a^2+b^2, we can now express the expression of the prime number of 1mod4 and the prime number of 3mod4, that is, all prime numbers. By the way, let's leave the prime number 2 alone.