Prime Factors:
The Royal Sevens
It is said by those who study such things that the number seven is the number of completion. This is, of course, based on the creation of the world being finished in six days and God resting on the seventh day. I accept this, for it was codified along with the fourth commandment on tablets of stone.
It is for this reason that I have decided to call the related primes ending in the number 7 the "Royal Family" of primes. Seven is a combination of three and the first square number, that is, the number 4.
However, the important feature is the relationship with the number 5. It just so happens that 7 is a "mirror image" of 3. Where 3 is two less that 5, 7 is two more. This translates into a chart very much like that of the threes.
However, in this case, subtraction back to the 5 will get the answer we need. I will start with 17 and backtrack back to 7.
What relationship does 1 have to 7 to get a something divisible by 17. It so happens that 5 times 7 is 35 and 2 times 17 is 34. That looks good, since they are just 1 apart.
The bigger number is the 35, so the operation will have to be subtraction. It will look odd, but the multiples in our present format will all be negative integers. But, then I know that -1 is just as good a factor as +1.
Here is the chart (with seven as "07" inserted)
07 => 0-(7*2) = 0-14 = -14
17 => 1-(7*5) = 1-35 = -34
37 => 3-(7*11)= 3-77 = -74
47 => 4-(7*14)= 4-98 = -94
67 => 6-(7*20)= 6-140 = -134
87 => 8-(7*26)= 8-186 = -174
Just like with the threes in the mirror, the multipliers increase by 3. In this case, they start with 2 instead of 1, so the "mirror" is a bit warped. The columns are not as neat as the numbers on the left don't match those on the right. We actually have to do some math to see what is happening here.
When you put a candidate for division in the left column the format makes more sense. However, with the multipliers getting into the twenties, we need to step back and concentrate on getting our math right.
Let's pick 37 as our prime. I will use a very simple multiple of 37 to illustrate the formula:
111 => 11-(1*11) = 11-11 = 0. As a matter of fact, 37 times 3 is 111.
To get a more obvious answer, let us use the square of 37.
1369 => 136-(9*11) = 136-99 = 37. Yep, 37 is a factor of 37!
One more thing, since 7 is 5+2, if you need to multiply something by 7, you only need to know the twos table and addition.
789 x 7 = ???
789 x 10 = 7890
7890/2 = 3945
789*2 = 1578
3945
+1578
5523
So, what do I know?
Because of the special relationship between 5 and 2, the tables for the 3 and 7 families look much alike. Practically mirror images, one finds a "positive" difference by adding, while the other uses the "negative" difference by subtracting. Each progresses by a factor of 3 between decades.
And so, the chart with only the primes:
07 => 0-(7*2) = 0 -14 = -14
17 => 1-(7*5) = 1 -35 = -34
37 => 3-(7*11)= 3 -77 = -74
47 => 4-(7*14)= 4 -98 = -94
67 => 6-(7*20)= 6-140 = -134
87 => 8-(7*26)= 8-186 = -174
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