Last night I decided to expand my number theory idea a little bit in Excel. Since the computer I’m borrowing is Windows 98, I’m not sure whether or not newer versions offer more size than this. But the parameters I was able to use are as follows:
Height of sheet = 65,536 cells
Width of sheet = 256 cells
Obviously, I would only be able to construct a square up to 256 x 256. However, I wanted to view as much as possible down into the 65,536 cells, so I decided to use all of the sheet.
As for method, in case you’re wondering, no I didn’t manually type in each box. Instead, what I did was copy the pattern and then paste it in the entire row. Unfortunately, if the box you select to paste the data into doesn’t evenly match a multiple of the box you first selected, Excel wouldn’t let you paste the data in. For example, if you want to paste in the multiples of 6, you can highlight the first 6 rows of the 6th column. When you highlight the entire column to paste it back, however, you need to ensure that the number of boxes you selected is a multiple of 6. In this instance, pasting into all 65,536 would give you an error.
Naturally, you can simply hold the Shift button and press “up” on the keyboard and try to paste it again until it finally accepts it when it gets to the right size (in this case, when you only highlight the first 65,532 cells). This works okay for the first few numbers, but once you get up into the teens and twenties (and beyond), it’s ridiculous trying to paste in that manner. Thankfully, there’s an easier manner.
Suppose you need to know what box column 98 would fit in. Using the default Microsoft calculator (in scientific mode) you can easily find this out by using the MOD function. This function will give you the remainder of x/y. So you simply type in 65536 [MOD] 98. This yields the remainder of 72. In the old method of trying each box one cell range smaller until it fits, you would have had to do this 72 times! But with this method, you can simply subtract 72 from 65536 and see that 65464 is the last cell that will fit. Highlight only to that cell, and paste in your data. Voila, mission accomplished.
Even with this easier method, it still takes quite some time to fill in the spreadsheet, so I only did the first 100 numbers last night. I’ll flesh out the rest later. But with the first 100 numbers in, some very interesting patterns are beginning to show up. Amazingly, the first one is found right at the end, starting at row 65,520.
65,520 is divisible by 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. It is likewise divisible by 12, 13, 14, 15, 16, 18, 20, 21, 24, 26, 28, 30, 35, 36, 39, 40, 42, 45, 48, 52, 56, 60, 63, 65, 70, 72, 78, 80, 84, 90, and 91. And that’s just the factors it has of the first 100 digits (for those counting, that’s 41 of the first 100 numbers that factor into the number 65,520: 41% of them!).
Amazingly, neither 65,519 nor 65,521 had any factors other than 1 between 1 and 100. It’s very possible that either of these numbers could be prime, although not proven (by me) yet.
65,520 is an obvious “spike” in the graphic. What’s amazing about these spikes is they occur in a regular pattern: every six rows.
The first number that has three straight factors in a row is the number 6, divisible by 1,2, and 3.
The first number that has four straight factors in a row is the number 12, divisible by 1, 2, 3, and 4.
The first number that has six straight factors in a row is the number 60, divisible by 1, 2, 3, 4, 5, and 6.
And so on. The number 360, for instance, is divisible by all the numbers between 1 and 10 except for 7. The first number that finally includes all of the first seven digits is 420.
And always, on either side of these spikes, there are voids, and often these voids are prime numbers. So we have the following:
5 (PRIME)
6 (SPIKE) 4 factors
7 (PRIME)
11 (PRIME)
12 (SPIKE) 6 factors
13 (PRIME)
17 (PRIME)
18 (SPIKE) 6 factors
19 (PRIME)
23 (PRIME)
24 (SPIKE) 8 factors
25 (LULL) 3 factors (1, 5, 25)
29 (PRIME)
30 (SPIKE) 8 factors
31 (PRIME)
35 (LULL) 4 factors (1, 5, 7, 35)
36 (SPIKE) 9 factors
37 (PRIME)
41 (PRIME)
42 (SPIKE) 8 factors
43 (PRIME)
47 (PRIME)
48 (SPIKE) 10 factors
49 (PRIME)
53 (PRIME)
54 (SPIKE) 8 factors
55 (LULL) 4 factors (1, 5, 11, 55)
59 (PRIME)
60 (SPIKE) 12 factors
61 (PRIME)
65 (LULL) 4 factors (1, 5, 13, 65)
66 (SPIKE) 8 factors
67 (PRIME)
71 (PRIME)
72 (SPIKE) 12 factors
73 (PRIME)
77 (LULL) 4 factors (1, 7, 11, 77)
78 (SPIKE) 8 factors
79 (PRIME)
[Interestingly, 80 also shows a spike here, despite not being a multiple of 6; but the pattern of 6s continues regardless, and the only number next to 80 that is prime is 79, which is also next to 78]
83 (PRIME)
84 (SPIKE) 12 factors
85 (LULL) 4 factors (1, 5, 17, 85)
89 (PRIME)
90 (SPIKE) 12 factors
91 (LULL) 4 factors (1, 7, 13, 91)
95 (LULL) 4 factors (1, 5, 19, 95)
96 (SPIKE) 12 factors
97 (PRIME)
101 (PRIME)
102 (SPIKE) 8 factors
103 (PRIME)
And there you go. The only primes not connected to a multiple of 6 between 1 and 103 are the early primes 2 and 3 (which multiply together to get 6—coincidence?)
Further, if you look at the factors involved in the numbers next to six that are NOT prime, you’ll see that all the other factors are prime numbers themselves! Thus, though 91 is not prime, it’s factors excluding 1 and itself (7 and 13) are both prime.
Of course, extending this further requires more math than my computer is currently able to handle :-( But perhaps there is a way to develop this theory further. Is it the case that all prime numbers are close to a multiple of 6? Is the pattern that 5 reoccurs next to 6 important (since any number other than 5 that ends in 5 will not be prime)? If we use a simple Y for “Yes, there is a multiple of 5 next to this multiple of 6″ and N for “No, there isn’t”), we have the following pattern for the multiples of 6:
Y N N Y N Y N N Y N (the period of this pattern is Y N N Y N; and also interestingly if anyone cares, I actually “hear” this period as a rhythmic pattern, in 4/4 time, of |:1 – 2 – 3 – 4&:| where 1 and 4 are the Y and 2, 3, and the & are N, and I accent the Y giving it a syncopated rhythm of Y N N YN Y N N, etc. Perhaps this is just more proof that I am odd….but then, only odd numbers can be prime, so take that!)
In any case, every time there is a Y there can be no more than 1 prime number next to the 6 since the other number is divisible by 5.
Furthermore, it would be much more interesting to see how this sequence carries out in higher numbers (alas, only being able to get to 256 is just WAY TOO LIMITING) to get to the interesting stuff. I can’t wait until I get my computer back, because I could program it to do this fairly easily instead of having to use Excel. Grrr.
Oh well. C’est la vie. For this, I blame Bush too.
