PRIME NUMBERS AND GOLDBACH'S CONJECTURE
By Philip Starmer
Charlotte, North Carolina 28270-1542
Email: st9m6@windstream.net
ABSTRACT
In 1742, Christian Goldbach wrote to Leonard Euler speculating that every even number greater than four can be expressed as the sum of two prime numbers, a prime number being such that it can only be divided by one and itself. Non-prime or composite numbers have various factors. While this conjecture has been shown to be correct up to very large numbers there has been no proof, so far, that it is always true, except when half the even number is a prime number. With the exception of 2, all even numbers are composite numbers. Odd numbers can be either prime or composite. While the number one is not considered to be a prime number, in this study it will used as a substitute for two since only odd numbers are considered. .
Odd numbers can be derived from one of three equations, y = 3 + 6x, y = 5 +6x and y = 7 + 6x, where x is an integer. While the first equation only gives composite numbers the other two give a mixture of composite and prime numbers. While prime numbers do not show any regularity composite numbers do. Prime numbers occur where there is a void in the composite network. For most, if not all, even numbers there are multiple prime-prime pairs. Using a simple example, it is seen that when an even number is not divisible by 3 the prime-prime pairs that survive a filtering process can in one case be defined by the second equation and in another by the third equation. When an even number is divisible by 3, the prime-prime pairs that survive have one member from the second equation and one from the third. This suggests that, since primes are not random but are controlled by the above equations, prime-prime pairs could survive at all values of even numbers. However, there is no way to prove or disprove this.
The number of primes up to N is defined as pi(N). This paper shows that for the number 2N the upper limit of prime-prime pairs is pi(2N) – pi(N) and the lower limit of prime-composite pairs is 2pi(N) – pi(2N). Graphs, derived from the data, suggest that the upper Iimit of prime-composite pairs is also pi(2N) – pi(N) which could mean that the lower limit of prime-prime pairs is the same as that of prime-composite pairs, namely 2pi(N) – pi(2N). However, an example is given where the lower limit for prime-prime pairs is less than 2pi(N) – pi(2N) which raises the possibility that it could, in some cases, be zero.
At this point it has not been possible to prove or disprove Goldbach's Conjecture.
KEY WORDS: GOLDBACH, CONJECTURE, PRIME, NUMBER, COMPOSITE, NON-PRIME, EULER, PI, ODD-NUMBER, EVEN-NUMBER.
PRIME NUMBERS AND GOLDBACH'S CONJECTURE
Whole numbers, or integers, are divided into composite and prime numbers. Composite, or non-prime, numbers can be either even or odd and have various factors besides one and the number itself. For example, 60 has the factors 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30 besides 1 and 60, which is why it was chosen by the Babylonians. Prime numbers are only divisible by one and the number itself. With the exception of 2, all prime numbers are odd. While the number one is not considered to be a prime, in this study it will be used in place of 2 since all the tables will be concerned with odd numbers. A useful table of primes less than 1,000 is given in Reference (1). It will be seen that prime numbers are not random but occur where there are voids in the composite network. Both Euclid and Euler (1) have proved that there are an infinite number of primes.
Odd numbers can be derived from one of the following three equations, where x is an integer:
.y = 3 + 6x (1)
.y = 5 + 6x (2)
.y = 7 + 6x (3)
While Equation (I) only gives composite numbers, Equations (2) and (3) give a mixture of composite and prime numbers. Table 1 shows the primes generated by Equation (2) for numbers less than 1,000. Table 2 gives the prime numbers generated by Equation (3) There are 86 primes in Table 1 and 80 in Table 2 which together with 2 and 3, which are not listed, gives a total of 168 less than 1,000. Comparing Tables I and 2, it is seen that twin primes occur when the same value of x appears in both tables. In Table 3 there are 34 twin primes less than 1,000 and Table 4 shows that there are four quadruple primes. On the other hand, there are runs of composite numbers. For example, between primes 113 and 127 there are six composite numbers, namely 115, 117, 119, 121, 123 and 125 which prevent the formation of prime-prime pairs.
Most, if not all, even numbers have multiple prime-prime pairs. The schematic in Table 5 will show how various pairs of odd numbers can occur. Let `a' be a composite number generated by Equation (1), `b' be a number from Equation (2) and `c' from Equation (3),In the upper rows the numbers increase by 2 while in the lower rows they decrease by 2. This means that the sum of the upper and lower numbers in each case add up to the same even number.
(Click on tables to see full size)
Situation A will produce the pairs aa, bc and cb. While aa pairs will only give composite-composite pairs, bc and cb will give composite-composite, prime-composite and prime-prime pairs. Situation B will produce the pairs ac, bb and ca with only bb pairs giving composite-composite, prime-composite and prime-prime pairs with the others giving prime-composite and composite-composite pairs. In situation C the order is reversed with only cc pairs giving the three possible combinations. If prime numbers were randomly placed it is extremely possible that a given arrangement would arise in which every prime number was paired with a composite one. However, prime numbers are not random but are structured according to mathematical equations, albeit simple ones, and fill in the voids in the composite structure. Therefore, it is tempting to suggest, but impossible to prove or disprove, that prime-prime pairs exist for all even numbers.
Four even numbers are shown in Table 6, two of which (996 and 1,002) are divisible by three and two (998 and 1,000) which are not. Situation A is represented by 996 and 1002 in which the prime-prime pairs consist of one number from Equation (2) and one number from Equation (3). Even numbers divisible by 3 have the highest number of prime-prime pairs since, as shown in Situation A above, the composite numbers are paired together and cancel one another. The number 1000 is representative of Situation B since all the primes from Equation (3) are aligned with composite numbers leaving Equation (2) as the source of prime-prime pairs. With 998, all the prime-prime pairs are based on Equation (3) since all the primes from Equation (2) are aligned with composite numbers that are multiples of 3 to give only prime-composite pairs. Situation C, as illustrated by 998, has the lowest number of prime-prime pairs since not only are composites based on 3, namely 3, 9, 15, 21 etc., act as a filter but also composites based on 5, namely 5, 15, 25, 35 etc, have the same effect.
In Table 7 there are columns headed by an ascending list of prime numbers:
1 3 5 7 11 ............................................................................m
Below are rows of odd numbers of the form:
.z y x w v ..............................................................................n
Such that z + 1 = 2N, y + 3 = 2N, x + 5 =2N................... m + n = 2n
Note, the row expands to another column each time n is a prime number. Also, when n is a prime number m and n are equal. In other words, 2N is the sum of two equal prime numbers. Therefore, Goldbach's Conjecture is true when N is a prime number. The number of primes up to, and including, N is defined as pi(N).
(click on table to see full size)
On the left side of Table 7 are three columns X, Y and Z. The column headed by X records the total number of pairs in which one number is a prime. This is equal to the number of primes at the head of the columns which in turn is equal to pi(N). These values are plotted in Figures 1 and 2 as an "x". Note that the values form a step-wise graph. A good approximation to pi(N) can be obtained from the Prime Number Theorem( 1) which is:
.pi(N) = N/InN (4)
The curves labeled "B" in Figures 1 and 2 were derived using Equation (4) and follow the general trend of the data. Curves labeled "A" were derived from the same equation using 2N instead of N.
Column Y gives the number of primes in a row and, since these are coupled with the primes at the head of columns, Y is the number of prime-prime pairs. These are plotted in Figure 1 as a "+". The number of primes from 1 to m and n to z is pi(2N) while the number of primes from 1 to m is pi(N). Thus, the highest number of primes in a row available to pair with primes at the head of a column is .pi(2N) — pi(N). Therefore,
The Upper Limit for prime-prime pairs = pi(2N) — pi(N) (5)
Curve C in Figure 1 is derived by subtracting Curve B from Curve A and describes the Upper Limit for prime-prime pairs with most values being lower than this curve.
For the numbers in Table 6, pi(I000) = 168 and pi(500) = 95 so the Upper Limit for prime-prime pairs is 73 with the highest actual value being 36.
Column Z gives the number of composites in a row and, thus, the number of prime-composite pairs. These are plotted in Figure 2 as an "o". Since the total number of primes is pi(N) and the Upper Limit of prime-prime pairs is . pi(2N) — pi(N) the Lower Limit of prime-composite is the difference so,
Lower Limit for prime-composite pairs = 2pi(N) — pi(2N) (6)
This is shown in Figure 2 as Curve D which is obtained by subtracting Curve C from Curve B. For completion, Curve D is drawn in Figure I and Curve C is drawn in Figure 2. The latter gives the impression that the Upper Limit for prime-composite pairs could be the same as that of prime-prime pairs. This suggests that the Lower Limit of prime-prime pairs could be same as that for prime-composite pairs, namely 2pi(N) — pi(2N). Using the above numbers for pi(1000) and pi(500) the Lower Limit is calculated to be 22 which greater than 18, the lowest value in Table 6, so the assumption is not valid.
(click table to see full size)
Curves C and D correspond to the Goldbach's Comet Curves of Fliegel and Robertson (3). As will be seen above, each even number has its own unique combination of prime-prime pairs which can not be predicted in any way. It is relevant to point out that for the even number 2N there are more primes in the interval 1 though N than there are in the interval N through 2N. The ratio is
Ratio = [pi(2N) — pi (N)]/pi(N) (7)
= [pi(2N)]/pi(N) - 1 (8)
Substituting the Prime Number Theorem from Equation (4)
Ratio = [21n(N)]/ ln(2N) - 1 (9)
Using this formula, for 2N = 100, the ratio is 0.74, for 1000 it is 0.8 and for a million it is 0.9. The actual figures for 100 and 1000 are 0.67 and 0.76, respectively. This means that there are many primes in both segments of the even number.
Figure 3 uses data from References (1) and (2) using logarithms to the base 10 (log), as opposed to natural logarithms (ln), since the values of N are given as powers of 10. This Figure shows that as the even number increases the number of primes less than that number also increases. At first, it was thought that this would be a simple way to calculate the number of primes less than a given number. However, as shown by Figure 4, an attempt to exactly fit the data was unsuccessful.
At this point it has not been possible to prove or disprove Goldbach's conjecture. However, it is relevant to point out that with even numbers not divisible by three, one third of the odd number pairs are a mixture of prime-prime, prime-composite and composite-composite pairs while with even numbers divisible by three the proportion is two thirds. This would imply that all even numbers have some prime-prime pairs which would support Goldbach. It is hoped that this study will stimulate a more sophisticated and aggressive approach to the problem.
REFERENCES
- John Derbyshire, Prime Obsession, 2001, Joseph Henry Press, 500 Fifth Street, N.W., Washington D.C. 20001 ISBN 0-309-08549-7
- Calvin C.Clawson, Mathematics Mysteries, the beauty and magic of numbers, Perseus Books, Cambridge, Massachusetts, 1999 ISBN 0-7382-0259-2
- Henry F.Fliegel and Douglas S.Robertson, Journal of Recreational Mathematics, Volume 21 No. 1, 1989 (from Reference 2)
Philip H. Starmer is a retired BF Goodrich Scientist. He lives and works in Charlotte, NC.
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