I had earlier written about prime numbers. Continuing from that post, I got interested in Goldbach's Conjecture.
The strong conjecture (SGC) holds that every even number can be expressed as a sum of two primes. The weak conjecture (WGC) holds that every odd number can be expressed as a sum of three primes - the primes need not be all different. As per the wiki link above, the WGC has been proved in the last few years while the SGC remains unproven even today.
If a and r are even positive numbers and p, q, s are prime numbers then any number a can be expressed as some p+q. (SGC)
And any odd number y can be expressed as the sum of three primes (=p+q+s). (WGC).
If SGC were true then to prove WGC:
We know that any odd number can be expressed as a sum of another smaller prime number and some even number. The latter even number can be expressed as the sum of two prime number (as per SGC).
So any odd number can be expressed as a sum of three prime numbers.
Coming to the proof of SGC: These are my thoughts. (No proof yet.)
a = p + q :
a+2 = p + q + 2 = (p + r + 2) + (q - r) where (p + r + 2) and (q - r) themselves are both positive and prime numbers. Note that the even number r can be positive or negative. But p+r+2 and q-r should both be positive. The minimum absolute value of r is the lower of (p+1) or (q-1).
Meaning from the values of p and q, there are two prime numbers which are r+2 higher than one number and r less than the other number. This predicts the proximity of a prime number from another prime.
How did we get p+1? The minimum value of p+r+2=1, so r = -p-1. Minimum absolute value of r = p+1.
This will be continued later.
Searching for prime numbers
I was thinking of the list of prime numbers. When you (a) go 2 to the left of or 4 to the right of any odd number in the number line you will likely get a prime number. If you find composite numbers instead, (b) go 4 to the left of and 2 to the right of the odd number. You will likely find prime numbers.
Example: start from 15. (a) 2 the left of 15 is 13. 4 to the right of 15 is 19. Both 13 and 19 are prime numbers. Using procedure (b) 2 to the right and 4 to the left of 15 are 17 and 11, which are also prime numbers
Start with 67. Procedure (a) 2 to the left of and 4 to the right of 67 are 65, 71, of which 71 is a prime but 65 is not. Procedure (b) 2 to the right and 4 to the left of 67 are 69 and 63 both of which are non primes.
Starting with the primes 5 and 7. Add 6 and multiples of 6 to each of these numbers and leave out the numbers that end in 5. The resultant numbers are likely to be primes (except the occasional multiples of 7, 11, 13 etc).
Thus 5+6x, 7+6y (where x, y are positive integers) provide better source of prime numbers. Of course, not all of these numbers will be prime as explained above. But, all prime number will definitely be part of one of these two series. When I say better I mean, the probability of a number in that series being prime is higher. The percentage of numbers that are primes in the two series above is about 38% as against about 10% when all positive numbers were considered. The series above are rich in prime numbers. Incidentally while browsing, I realized that 5+6x and 7+6y can be written in a simpler way as 6n+1 and 6n-1.
I generated the series above as a set of potential prime number candidates. I determined whether each candidate has any divisors. I used the same prime candidates from the series above as divisors. Remember that composite numbers obviously can never be divisors for prime numbers. I do not mean prime numbers will have other prime numbers as divisors. When you have a number a prime number as a potential candidate - where we do not know yet if the number is a prime, we need to only see if the candidate has any primes as divisors.
I found an interesting thing.
Prime numbers were about 23% of all numbers till the first 100 (my first prime number was 5, I left out 2 and 3). The ratio decreased rapidly first and slowly subsequently till it reached 10.3% for numbers up to 50000.
There is a theorem that states that the prime number % upto the number x = 1/ln(x) as x becomes very large.
Estimation of prime numbers through factors is continued in this post: http://vbala99.blogspot.com/2016/05/estimating-fraction-of-numbers-that-are.html
Coming to the proof of SGC: These are my thoughts. (No proof yet.)
a = p + q :
Note that p, q are prime numbers. We do not know whether p is bigger than or smaller than or equal to q.Assume above equation is true for any even number a. Take the next higher even number a+2. We have to prove that there exists some even number r such that,
a+2 = p + q + 2 = (p + r + 2) + (q - r) where (p + r + 2) and (q - r) themselves are both positive and prime numbers. Note that the even number r can be positive or negative. But p+r+2 and q-r should both be positive. The minimum absolute value of r is the lower of (p+1) or (q-1).
Meaning from the values of p and q, there are two prime numbers which are r+2 higher than one number and r less than the other number. This predicts the proximity of a prime number from another prime.
This will be continued later.
Searching for prime numbers
I was thinking of the list of prime numbers. When you (a) go 2 to the left of or 4 to the right of any odd number in the number line you will likely get a prime number. If you find composite numbers instead, (b) go 4 to the left of and 2 to the right of the odd number. You will likely find prime numbers.
Example: start from 15. (a) 2 the left of 15 is 13. 4 to the right of 15 is 19. Both 13 and 19 are prime numbers. Using procedure (b) 2 to the right and 4 to the left of 15 are 17 and 11, which are also prime numbers
Start with 67. Procedure (a) 2 to the left of and 4 to the right of 67 are 65, 71, of which 71 is a prime but 65 is not. Procedure (b) 2 to the right and 4 to the left of 67 are 69 and 63 both of which are non primes.
Starting with the primes 5 and 7. Add 6 and multiples of 6 to each of these numbers and leave out the numbers that end in 5. The resultant numbers are likely to be primes (except the occasional multiples of 7, 11, 13 etc).
Thus 5+6x, 7+6y (where x, y are positive integers) provide better source of prime numbers. Of course, not all of these numbers will be prime as explained above. But, all prime number will definitely be part of one of these two series. When I say better I mean, the probability of a number in that series being prime is higher. The percentage of numbers that are primes in the two series above is about 38% as against about 10% when all positive numbers were considered. The series above are rich in prime numbers. Incidentally while browsing, I realized that 5+6x and 7+6y can be written in a simpler way as 6n+1 and 6n-1.
I generated the series above as a set of potential prime number candidates. I determined whether each candidate has any divisors. I used the same prime candidates from the series above as divisors. Remember that composite numbers obviously can never be divisors for prime numbers. I do not mean prime numbers will have other prime numbers as divisors. When you have a number a prime number as a potential candidate - where we do not know yet if the number is a prime, we need to only see if the candidate has any primes as divisors.
I found an interesting thing.
Prime numbers were about 23% of all numbers till the first 100 (my first prime number was 5, I left out 2 and 3). The ratio decreased rapidly first and slowly subsequently till it reached 10.3% for numbers up to 50000.
There is a theorem that states that the prime number % upto the number x = 1/ln(x) as x becomes very large.
Ln denotes log to the base e where e = 2.71828 approximately. The exact definition formula for e is (1+1/n)^n as n tends to infinity.Incidentally 1/(log 50000 to the base e)= 9.2%, which kinda matches with the figure of 10.3% that I found.
Estimation of prime numbers through factors is continued in this post: http://vbala99.blogspot.com/2016/05/estimating-fraction-of-numbers-that-are.html
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