[This article was sent in by Mike Fink, owner of Domino Cubes, a Z88 Dealer. Once Mike told me about his work, I thought this would be a good article. Plus, Mike has had problems getting Academia to look at his work. I plan to additionally post this article to one of the math newsgroups.
So, look over this article. I hope you understand it better than I (just a bit over my head). Pass it to others that might be interested. I believe that it needs some peer review to bolster it.
The article appears just as I received it. Any typo’s are purely mine. Anyone seriously interested in the article can send me a note and I’ll send a hardcopy of the original article (just in case I mistyped something). – Ed]
I have spent 27 years of my life searching for the answer to the question, is there a formula that will generate only prime numbers or is that impossible? Many people since time began sought the answer and none have found it; which is why I began the search. In my warped sense of my ego – at 17 years of age, I thought I could do anything since I had never failed at anything yet at that age.
For the rest of this article the character E will represent the Sigma (Sumation) math symbol. Also, a ‘ placed before a character means that that character is subscripted. For example, V’1 means V sub 1.
- All primes (except 2, 3, and 5) end in 1, 3, 7 or 9.
- All primes exist in one of 24 families.
- The 24 families are the result of my research.
- The similarities in the 4 building blocks of DNA and RNA plus the 24 chromosomes of man, to the 4 building blocks of primes and its 24 families, has led me to conclude that there exists a definite relationship to the secrets of primes and the secrets of double spiral helices.
- I am not knowledgeable enough to enter the field of bio-genetics, and maybe I am not smart enough anymore to find the connection. I am convinced that my conjecture as to this relationship is of such importance, that someone in the bio-genetic-math community should pursue this premise.
The following article gives the framework of my results, but first let me explain a few things;
- The level of mathematics required to understand my statements, requires ordinary high school stuff.
- The level of logic and reasoning is a little harder.
- All numbers divisible by 2, 3, and 5 have been eliminated from the V numbers generated by EQ. 2, which makes the sample to investigate much smaller.
- The composites are always made from 2 or more primes.
- Composites that are squares can only occur in V’a numbers ending in 1 or 9.
PREMISE: A formula (or formulae) which will allow a number N to be investigated to determine if it is a prime – is possible.
ATTITUDE: If a formula is found that generates all primes but contains a few non-primes that smell differently, and a test to remove the smelly ones is an easy test, this would and should satisfy the conditions for a viable solution. What remains then is a definition for easy.
- All primes beyond 2, 3, and 5 have the units digits of 1, 2, 7, or 9.
- EQ. 2 generates all primes, plus a small group of non-primes and 1 is considered prime for purposes of family identifications, or maybe it is a prime.
- In EQ. 2, all primes have the factors or roots of R’1=1 and R’2=N; for the non-primes R’1=7 or larger in the form of R’1=V+90C (See EQ. 7 & 8).
- E’m signifies the summing of the digits in a number to a single digit, this digit, for numbers generated by EQ. 2, can only be 1, 4, 7, 2, 5 or 8 (6 choices).
- POSTULATE: E’m of N = (E’mR’1)(E’mR’2) = (E’mV’1)(E’mV’2).
- POSTULATE: All numbers generated by EQ. 2 fall into 24 families, ie: E’m has only 6 possibilities (1,4,7,2,5,8) and the units digit of N only has 4 ways to happen (1,3,7,9) creating (6×4) 24 families.
- The first 12 numbers generated by EQ. 3 and EQ. 4 (See Table 1), yield “V” numbers; 12 V’a and 12 V’b.
- HYPOTHESIS: To find a formula to satisfy the premise considering all potential primes at once is difficult, if not impossible; but by breaking the potential primes into the 24 families, order can be found within each family.
- AXIOM:
A. All numbers generated by EQ. 2 are either prime or not.
B. All non-primes generated by EQ. 2 have roots that can be found by EQ.’s 16, 17, and 18 — or EQ.’s 9 and 10.
C. Therefore if B is true, all other number are prime. - PROOF THAT B IS TRUE:
A. Table 2 shows that all the combinations of (V’1)(V’2) that will satisfy postulate 5.
B. For any N in a given “V” family, there are either 12 or 14 combinations of (V’1)(V’2) to be inserted into EQ.’s 16, 17, and 18.
C. If N is not prime, integer roots other than R’1=1 will appear: therefore B of 9 is true.
EQ. 1: N = V+90C (See EQ. 5)
EQ. 2: N = 6n+-5 where n<>5K 1<=k<=infinity
EQ. 3: N'a = 6n-5 (See Table 1)
EQ. 4: N'b = 6n+5 (See Table 1)
EQ. 5: N'a = V'a+90C'a
N'b = V'b+90C'b (See EQ. 1)
EQ. 6: C = (N-V)/90 (See EQ. 1)
EQ. 7: N = R'1R'2 (R'1 and R'2 = Roots or Factors)
EQ. 8: N = (V'1 + 90C'1)(V'2 + 90C'2) = V+90C
EQ. 9: N'a = X^2 - 9Y^2
EQ. 10: N'b = 9X^2 - Y^2
EQ. 11: N'aN'a1 = N'a2
N'bN'b1 = N'a3
N'aN'b = N'b2
EQ. 12: X'a = (R'1 + R'2)/2 = 3(n'1+n'2)+-5
EQ. 13: X'b = (R'1 + R'2)/6 = (n'1 + n'2)
EQ. 14: Y'a = (R'2 - R'1)/6 = (n'2 - n'1)
EQ. 15: Y'b = (R'2 - R'1)/2 = 3(n'2 - n'1)+-5
EQ. 16: If C'1 = 0, (N-V'1V'2)/V'1 = 90C'2
EQ. 17: If C'2 = 0, (N-V'1V'2)/V'2 = 90C'1 (See EQ. 8)
EQ. 18: When C'1 or C'2 <>0;
(N-V'1V'2-90V'2C'1)/(V'1+90C'1) = 90C'2
(N-V'1V'2-90V'1C'2)/(V'2+90C'2) = 90C'1
Table 1
| n | V’a | V’b |
|---|---|---|
| 1 | 1 | 11 |
| 2 | 7 | 17 |
| 3 | 13 | 23 |
| 4 | 19 | 29 |
| 6 | 31 | 41 |
| 7 | 37 | 47 |
| 8 | 43 | 53 |
| 9 | 49 | 59 |
| 11 | 61 | 71 |
| 12 | 67 | 77 |
| 13 | 73 | 83 |
| 14 | 79 | 89 |
TABLE 2 VALUES FOR V’2
-------------------------------------------------------------------------
| V'a 1 V'b | V'a 3 V'b | V'a 7 V'b | V'a 9 V'b |
-------------------------------------------------------------------------
| 1| 4| 7| 2| 5| 8| 1| 4| 7| 2| 5| 8| 1| 4| 7| 2| 5| 8| 1| 4| 7| 2| 5| 8| EmR'2|
| 01|91 31 61 11 41 71|73 13 43 83 23 53|37 67 07 47 77 17|19 49 79 29 59 89|
| 73|37 67 07 47 77 17|91 31 61 11 41 71|19 49 79 29 59 89|73 13 43 83 23 53| 1|
| 37|73 13 43 83 23 53|19 49 79 29 59 89|91 31 61 11 41 71|37 67 07 47 77 17| EmR'1|
| 19|19 49 79 29 59 89|37 67 07 47 77 17|73 13 43 83 23 53|91 31 61 11 41 71|
| 31|61 91 31 41 71 11|43 73 13 23 53 83|07 37 67 77 17 47|79 19 49 59 89 29|
| 13|07 37 67 77 17 47|61 91 31 41 71 11|79 19 49 59 89 29|43 73 13 23 53 83| 4|
| 67|43 73 13 23 53 83|79 19 49 59 89 29|61 91 31 41 71 11|07 37 67 77 17 47| EmR'1|
| 49|79 19 49 59 89 29|07 37 67 77 17 47|43 73 13 23 53 83|61 91 31 41 71 11|
| 61|31 61 91 71 11 41|13 43 73 53 83 23|67 07 37 17 47 77|49 79 19 89 29 59|
| 43|67 07 37 17 47 77|31 61 91 71 11 41|49 79 19 89 29 59|13 43 73 53 83 23| 7|
| 07|13 43 73 53 83 23|49 79 19 89 29 59|31 61 91 71 11 41|67 07 37 17 47 77| EmR'1|
| 79|49 79 19 89 29 59|67 07 37 17 47 77|13 43 73 53 83 23|31 61 91 71 11 41|
| 11|41 11 71 91 61 31|23 83 53 73 43 13|77 47 17 37 07 67|59 29 89 19 79 49|
| 83|77 47 17 37 07 67|41 11 71 91 61 31|59 29 89 19 79 49|23 83 53 73 43 13| 2|
| 47|23 83 53 73 43 13|59 29 89 19 79 49|41 11 71 91 61 31|77 47 17 37 07 67| EmR'1|
| 29|59 29 89 19 79 49|77 47 17 37 07 67|23 83 53 73 43 13|41 11 71 91 61 31|
| 41|11 71 41 31 91 61|83 53 23 13 73 43|47 17 77 67 37 07|29 89 59 49 19 79|
| 23|47 17 77 67 37 07|11 71 41 31 91 61|29 89 59 49 19 79|83 53 23 13 73 43| 5|
| 77|83 53 23 13 73 43|29 89 59 49 19 79|11 71 41 31 91 61|47 17 77 67 37 07| EmR'1|
| 59|29 89 59 49 19 79|47 17 77 67 37 07|83 53 23 13 73 43|11 71 41 31 91 61|
| 71|71 41 11 61 31 91|53 23 83 43 13 73|17 77 47 07 67 37|89 59 29 79 49 19|
| 53|17 77 47 07 67 37|71 41 11 61 31 91|89 59 29 79 49 19|53 23 83 43 13 73| 8|
| 17|53 23 83 43 13 73|89 59 29 79 49 19|71 41 11 61 31 91|17 77 47 07 67 37| EmR'1|
| 89|89 59 29 79 49 19|17 77 47 07 67 37|53 23 83 43 13 73|71 41 11 61 31 91|
**|14 14 14 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 14 14 14 12 12 12|
-------------------------------------------------------------------------
** Possible Pairs
Left-most column contains V'1 numbers.
Mike Fink’s Remarkable Method of Finding Primes Without Dividing!!
- Choose any family; N=V+90C, say, N’b=V’b+90C, (EQ. 8)
- The largest composite number in that family is always known.
- Choose N’be larger then N’b.
- Is N’be a prime?
- The biggest composite less than N’be is known, ie; N’b.
- The smallest composite bigger than N’b can easily be generated; see example below.
- Every number in this family between these two non-primes, is, buy definition, a prime.
EXAMPLE: N’b = 77 + 90C = 707 = 77+90(7).. Largest known non-prime.
Let N’be = 1067; C’e = 11. Is 1067 a prime?
All non-primes, in this family, have roots = R’1R’2 = N’axN’bx = (V’ax+90C’ax)(V’bx+90C’bx)
R’1 must be equal to or greater than 7; See Tables 1 or 2.
(V'ax+90C'x)(V'bx+90Cx) = R'1 R'2 = N'bx
-------------------------------------------------
(91+90C'x) (77+90C'x) = (91) (77) = 7007
(07+90C'x) (11+90C'x) = (97) (11) = 1067 **
(13+90C'x) (89+90C'x) = (13) (89) = 1157
(19+90C'x) (23+90C'x) = (19) (113) = 2147
(31+90C'x) (17+90C'x) = (121) (17) = 2057
(37+90C'x) (41+90C'x) = (37) (41) = 1517
(43+90C'x) (29+90C'x) = (43) (29) = 1247
(49+90C'x) (53+90C'x) = (49) (53) = 2597
(61+90C'x) (47+90C'x) = (61) (47) = 2867
(67+90C'x) (71+90C'x) = (67) (71) = 4757
(73+90C'x) (59+90C'x) = (73) (59) = 4307
(79+90C'x) (83+90C'x) = (79) (83) = 6557
** = The smallest non-prime bigger than 707 and less than or equal to 1067. Therefore ALL family numbers between 707 and 1067 must be, and are PRIMES!
797, 887, and 977 are primes!
The numbers used above are from Table 2 and show the 12 possible V numbers that will satisfy Eq. 11 and postulate 5. The C’x numbers are chosen to be in the range of the problem.
It is now obvious that it is possible to deduce multiple primes without dividing, for each and every family. The time and effort required is less than for any other known method! Some families have 14 combinatorial pairs Em, the rest have 12.
[This is a almost literal translation of the original Z88 program by Mike. I’ve tested it with the example runs provided by Mike – Ed ]100 REMark Prime Numbers - Mike Fink
110 REMark This program ascertains the primality of any
120 REMark number less than a billion.
130 REMark (C) 1988 by Mike Fink with help from
140 REMark Robert Fink
150 INPUT " Enter Number ";n
160 dv = 0 : time = DATE
170 IF n = 2 OR n = 3 OR n = 5 THEN GO TO 360
180 IF n/2=INT(n/2) OR n/3=INT(n/3) OR n/5=INT(n/5) THEN GO TO 400
190 DIM x(24)
200 g = SQRT(n)
210 x(1)=7 : x(2)=11 : x(3)=13 : x(4)=17 : x(5)=19
220 x(6)=23 : x(7)=29 : x(8)=31 : x(9)=37 : x(10)=41
230 x(11)=43 : x(12)=47 : x(13)=49 : x(14)=53 : x(15)=59
240 x(16)=61 : x(17)=67 : x(18)=71 : x(19)=73 : x(20)=77
250 x(21)=79 : x(22)=83 : x(23)=89 : x(24)=91
260 c = 0
270 FOR y = 1 TO 24
280 v1 = x(y) + 90 * c
290 IF v1 > g THEN GO TO 360
300 v2 = n/v1
310 dv = dv + (24 * c) + y - 1
320 IF v2 = INT(v2) THEN PRINT " v1 = ";v1;" v2 = ";v2 : RUN
330 NEXT y
340 c = c + 1
350 GO TO 270
360 PRINT " N = ";n;" is a prime."
370 PRINT " No. of Divisions = ";dv
380 PRINT " Time = ";DATE-time
390 RUN
400 PRINT "N = ";n;" is not a prime"
410 RUN
Example Output:
N = 117 Is not prime.
N = 119 V'1 = 7 V'2 = 17
N = 1119 Is not prime.
N = 11119 Is a prime! No. of Divisions = 351
N = 966666661 Is a prime! No. of Divisions = 34,357,905