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Appendix II.   Significance (or Insignificance) of Results




The Bible-calculator, as a research tool, can provide much information on time intervals and their mathematical building blocks.  Yet as is typical in the "information age", more information requires more effort to sift, sort, and analyze it.  Patience, prudence, prayer, and guidance from teachers is required in deciding which results are significant, and which results are insignificant or misleading.

One trap the Bible student may face is over-excitement at finding a time interval (such as a dayspan) whose total breaks down into Biblically significant primes.  Such "impressive" results can be more plentiful than one would think.

This webpage is not an exhaustive mathematical education, but merely a quick look at how one must maintain a healthy skepticism and be ready to seek mathematical and statistical explanations.


Test:  How Commonplace are Primes of Biblical Interest?
{ 2, 3, 5, 7, 11, 13, 17, 23, 37, 43 }

Below:  An informal test of (admittedly short) dayspans from January 1 (in a non-leap year) to each day in June.  Each date-interval had the option to be recorded with the "inclusive" option, depending on which result (inclusive or not) produced smaller, or more Biblically significant, primes.

Those whose "factors" are limited to the primes { 2, 3, 5, 7, 11, 13, 17, 23, 37, 43 } are marked "Y"; the others, "N".

From 1-1 to:

N 6- 1 = 152 incl = 2^3 x 19
Y 6- 2 = 153 incl = 3^2 x 17
Y 6- 3 = 154 incl = 2 x 7 x 11
Y 6- 4 = 154 = 2 x 7 x 11
Y 6- 5 = 156 incl = 2^2 x 3 x 13
Y 6- 6 = 156 incl = 2^2 x 3 x 13
N 6- 7 = 158 incl = 2 x 79
N 6- 8 = 159 incl = 3 x 53
Y 6- 9 = 160 incl = 10 x 2^4
Y 6-10 = 160 = 10 x 2^4
Y 6-11 = 161 = 7 x 23
Y 6-12 = 162 = 2 x 3^4
N 6-13 = 164 incl = 2^2 x 41
Y 6-14 = 165 incl = 3 x 5 x 11
Y 6-15 = 165 = 3 x 5 x 11
N 6-16 = 166 = 2 x 83
Y 6-17 = 168 incl = 2^3 x 3 x 7
Y 6-18 = 169 incl = 13^2
Y 6-19 = 169 = 13^2
Y 6-20 = 170 = 10 x 17
Y 6-21 = 172 incl = 2^2 x 43
Y 6-22 = 172 = 2^2 x 43
N 6-23 = 174 incl = 2 x 3 x 29
Y 6-24 = 175 incl = 5^2 x 7
Y 6-25 = 176 incl = 2^4 x 11
Y 6-26 = 176 = 2^4 x 11
N 6-27 = 177 = 3 x 59
N 6-28 = 178 = 2 x 89
Y 6-29 = 180 incl = 10 x 2 x 3^2
Y 6-30 = 180 = 10 x 2 x 3^2

Results:  Out of 30 dayspans, 22 have divisors limited to the afore-mentioned set of interesting primes.

Perhaps a fairer test would be a similar set of long-term consecutive dayspans:

From 1-1-1 AD to:

N 6- 1-2001 = 730636 = 2^2 x 182659
N 6- 2-2001 = 730637 = 730637
N 6- 3-2001 = 730638 = 2 x 3^2 x 40591
N 6- 4-2001 = 730639 = 7^2 x 13 x 31 x 37
N 6- 5-2001 = 730640 = 10 x 2^3 x 9133
N 6- 6-2001 = 730641 = 3 x 23 x 10589
N 6- 7-2001 = 730642 = 2 x 11 x 33211
N 6- 8-2001 = 730643 = 17 x 42979
N 6- 9-2001 = 730644 = 2^2 x 3 x 60887
N 6-10-2001 = 730645 = 5 x 19 x 7691
N 6-11-2001 = 730646 = 2 x 7 x 52189
N 6-12-2001 = 730647 = 3^3 x 27061
N 6-13-2001 = 730648 = 2^3 x 91331
N 6-14-2001 = 730649 = 83 x 8803
N 6-15-2001 = 730650 = 10 x 3 x 5 x 4871
N 6-16-2001 = 730651 = 257 x 2843
N 6-17-2001 = 730652 = 2^2 x 13 x 14051
N 6-18-2001 = 730653 = 3 x 7 x 11 x 3163
N 6-19-2001 = 730654 = 2 x 365327
N 6-20-2001 = 730655 = 5 x 29 x 5039
N 6-21-2001 = 730656 = 2^5 x 3^2 x 43 x 59
N 6-22-2001 = 730657 = 73 x 10009
N 6-23-2001 = 730658 = 2 x 53 x 61 x 113
N 6-24-2001 = 730659 = 3 x 243553
N 6-25-2001 = 730660 = 10 x 2 x 7 x 17 x 307
N 6-26-2001 = 730661 = 41 x 71 x 251
N 6-27-2001 = 730662 = 2 x 3 x 47 x 2591
N 6-28-2001 = 730663 = 730663
N 6-29-2001 = 730664 = 2^3 x 11 x 19^2 x 23
N 6-30-2001 = 730665 = 3^2 x 5 x 13 x 1249

Remarks:  Out of 30 dayspans, 0 have divisors limited to the afore-mentioned set of interesting primes { 2, 3, 5, 7, 11, 13, 17, 23, 37, 43 }.  The reason this second example has fewer such entries is that, while our "favorite" primes still occur at predictable rates (explained below), the higher daycounts show the effects of a lengthy, steady accumulation of new and greater prime factors.

The continual, cyclical reappearance of these large factors creates a background noise of non-interesting primes that litters the field; a dense jungle of lesser-significance numbers through which shine the even rarer light shafts of purely "interesting" primes such as [772497 = 3^5 x 11 x 17^2].

With these larger dayspan amounts (730636 to 730665), there come many higher prime factors.  Yet all factors are, by definition, smaller than the number being analyzed (in this case, "dayspan").  The upward limit of a factor is (dayspan / 2).  Thus no factor in the above 30-day chart may exceed (730665 / 2), or 365332 (using integer division where any remainder-- in this case 1 -- gets discarded).

To see how this is true, look at the entry for 6-19 above.  As just stated, the highest possible factor for a number is (number / 2).  Here, 365327 x 2 = 730654.  Were any number higher than 365327 considered as a possible factor, a problem arises.

Imagine you have 730,654 cookies, and your goal is to divide them into equal parts.  If you take any number greater than one-half (365,327)-- let's say 370,000-- and put them into one pile, then there are not cookies enough left among the remainder (360,654) to make a pile equal to the first.


The Predictability of Occurrence of Prime Factors

Notice the predictable rate of appearance of all factors in the above examples:  For instance, the number 7 appears as a factor every seven days.  The number 13 appears as a factor of 730639 on 6-4, then once again on 6-17, 13 days later.  In predictable fashion, it appears 13 days later on 6-30.

What is happening is that on 6-4, the dayspan (730639) is expressable as (13 x 56203).  Think of this as "56,203 thirteens".  The number thirteen cannot appear again as a factor until the next batch of thirteen "ones" has been accomplished.  So 13 days later on 6-17, we have (730652) days, which is (13 x 56204), or, one more "thirteen" than you had before.

Look at the number 11 on 6-7.  It will appear again in the month of June, because in the 23 remaining days of June there is room for another eleven.  Since 11 <= 23, it will appear again.

Notice the number 29 on 6-20.  Because 29 < 30, it will appear at least once... but as there are not 29 days on either side of 6-20 that fall in June, it cannot appear a second time.

We can postulate that for any factor f in a sample-set R of size s, (where R is a list of factors for consecutive, non-zero numbers):

1.  If f <= s, then f must appear in R

2.  If f <= (s / 2), then f must appear in R more than once

3.  f will appear in R at most  (s + (f - 1)) / f  times (using integer division-- i.e., remainder is discarded)

4.  Where x = the number of elements at start of list R not containing f as a factor, f will appear in R exactly  (s + (f - 1) - x) / f  times (using integer division-- i.e., remainder is discarded)

Example:  


  s = 10,

  R = 
    { 21 ... 3 x 7 
      22 ... 2 x 11
      23 ... 23
      24 ... 2^3 x 3  
      25 ... 5^2
      26 ... 2 x 13 
      27 ... 3^3 
      28 ... 2^2 x 7
      29 ... 29
      30 ... 2 x 3 x 5 }
Let us test the above rules....

Test:  f = 3

By rule 1:   3 <= 10,  therefore 3 must appear as a factor in R

    TRUE.

By rule 2:   3 <= (10/2)   3 <= 5,  therefore 3 must appear as a factor in more than one element of R

    TRUE.  3 appears as a factor in four elements of R: {21, 24, 27, 30}

By rule 3:   (10 + (3 - 1)) / 3    (10 + 3) / 3    12 / 3 = 4, therefore 3 will appear as a factor in (at most) four elements of R

    TRUE.  3 appears as a factor in four elements of R: {21, 24, 27, 30}

To check rule 4, we choose a number for f that does not immediately appear as a factor:  namely, 4.   (10 + (4 - 1) - 3)) / 4   ((10 + 3) - 3) / 4   10 / 4 = 2 (remainder discarded), therefore 4 will appear as a factor in exactly two elements of R

    TRUE.  4 appears as a factor in two elements of R: {24, 28}  (note-- 2^2 = 4)


Yearspan Example:  Hazards / Inclusive Option

To be fair, we will look at numbers between the extremes of the two dayspan-charts above, the one having numerous interesting primes, and the other lacking any such sets at all.  A set of yearspans has numbers 1/365th the size of corresponding dayspans.  How common are interesting primes here?

Those whose "factors" are limited to the primes { 2, 3, 5, 7, 11, 13, 17, 23, 37, 43 } are marked "Y"; the others, "N".

Some Random, Consecutive Year-Spans and Their Factors 

Year 273 AD to:

Y  1901 AD = 1628      =  2^2   x   11   x   37
 N 1902    = 1629      =  3^2   x   181
 N 1903    = 1630      =  10   x   163
Y  1904    = 1632 incl =  2^5   x   3   x   17
Y  1905    = 1632      =  2^5   x   3   x   17
 N 1906    = 1633      =  23   x   71
 N 1907    = 1634      =  2   x   19   x   43
 N 1908    = 1635      =  3   x   5   x   109
 N 1909    = 1636      =  2^2   x   409
Y  1910    = 1638 incl =  2   x   3^2   x   7   x   13
Y  1911    = 1638      =  2   x   3^2   x   7   x   13
 N 1912    = 1639      =  11   x   149
 N 1913    = 1640      =  10   x   2^2   x   41
 N 1914    = 1641      =  3   x   547
 N 1915    = 1642      =  2   x   821
 N 1916    = 1643      =  31   x   53
 N 1917    = 1644      =  2^2   x   3   x   137
 N 1918    = 1645      =  5   x   7   x   47
 N 1919    = 1646      =  2   x   823
 N 1920    = 1647      =  3^3   x   61
 N 1921    = 1648      =  2^4   x   103
Y  1922    = 1650 incl =  10   x   3   x   5   x   11
Y  1923    = 1650      =  10   x   3   x   5   x   11
 N 1924    = 1651      =  13   x   127
 N 1925    = 1652      =  2^2   x   7   x   59
 N 1926    = 1653      =  3   x   19   x   29
 N 1927    = 1654      =  2   x   827
Y  1928    = 1656 incl =  2^3   x   3^2   x   23 [= 6 X 276]
Y  1929    = 1656      =  2^3   x   3^2   x   23 [= 6 X 276]
 N 1930    = 1657      =  1657

We see here that the primes 2, 3, 5, 7, 11, 13, 17, 23, 37, and 43 are not uncommon factors.  And yearspan factor-sets composed solely of these numbers are, while rarer, still not remarkably uncommon.  Five of the above yearspans qualify outright, as do an extra four thanks to the "inclusive" option.

The ability to select from inclusive/non-inclusive factor-sets is indeed legitimate, yet it can help stack the deck with "interesting" results.  With this option, the rate of natural occurrence can be doubled, as every day-/yearspan is given a second shot at containing desirable factors, when the end-day (or year) is "included" into the total span-- providing a completely new set of factors.

Example:

Say we are checking the dayspan 1-1 to 8-26 for significant primes:

1-1 to 8-26 not incl = 237 = 3 x 79

The 3 might be significant, but the 79 throws doubt on this.

So we try the "inclusive" option, which adds 1 to the daycount, by including both start and endpoints in the tally of days:

1-1 to 8-26 incl = 238 = 2 x 7 x 17

Now we have what might be tantalizingly significant results, as 2 and 7 bear significance, as does the comparatively rarer 17.  The 17 is interesting, but were we looking for a 17?  It is much easier to bump into interesting primes if we do not care which one(s), than it is to find a specific, single number.

* [Inclusive option, description and justification: Consider a timespan from Noon on a Sunday to Noon the next Sunday.  The dayspan is exactly 7 (i.e., the end is seven days after the start), yet the number of days involved is 8.  There are eight "dates" in effect during, or touched by, the 7-day span.  Sun - Sat (7 dates) plus the final Sunday (the 8th date).  Both inclusive and non-inclusive are valid methods of counting days in a dayspan.  The method which yields the more interesting of two factor-sets can freely be chosen, as either is valid.]


False Significance:  Examine the Mechanics of What is Going On

Occasionally we find a time "relationship" of seeming significance that can be dismantled by a closer look at the mechanics involved.

Kevin was born in 1963, and expects to be 48 in 2011.   [63, 48, 11]
Israel was reborn in 1948, and will be 63 in 2011.   [48, 63, 11]
Is there something going on here?  Some mystical alignment between 11, 48, and 63?  A mathematical trick?
Does this setup work for other number combinations?

The answer is simple.  This is an example of the commutative property of addition, taught in grade school:

A + B  =  B + A

  ...or...

If  A + B  =  C,  then  B + A  =  C

Convert the years to "years since 1900".  (While not a necessary step, it will serve to simplify the demonstration.)


1948 -->  48
1963 -->  63
2011 --> 111

48 (Israel reborn) + 63 (years later) = 111
63 (Kevin born)    + 48 (years later) = 111

48 + 63 = 111
63 + 48 = 111

Any number of values can be used in similar fashion.

John O'Leary / Biblecalculator.com