Compositions of Primes – A number’s fingerprint

Every number has the interesting property of being composed of exactly one set of lowest prime numbers multiplied together. Which is to say, 2*3 is 6, and there is no other way to make 6 than to multiply together 2 and 3. Anything else you used to make 6, would really just be other numbers that themselves were composed of 2 and 3. Though as 6 only has two factors, it doesn’t really demonstrate that very well.

Ten is made up of two primes, 2*5. The only way to make ten is to combine 2 and 5. The only thing that can go into ten is a series of five twos, or a series of two fives. This makes 2, 4, 5, 6, 8, rather special numbers when we’re dealing with a system where ten is the base. That’s because when we cross over the threshold into the next set of 10, those numbers stay the same, and continue to mean the same things, so long as those things can be said with only one 2, and one 5. That’s why any number ending in 2, 4, 6, and 8 , and 0 is divisible by 2, and any number ending in 5 and 0 is divisible by 5.

Twelve is a number with 3 compositional primes, or rather, prime factors. 2*2*3. It’s impossible to make twelve out of prime numbers that do not consist of two 2s, and one 3. However, twelve can also be written as 4*3. This is because 4 is made up of 2*2. And 2*2 and 3 make twelve. It’s also possible to make twelve as 6*2, because 6 is 2*3. It is however impossible to make twelve out of any number that is not made up of a 2, or a 3. It’s also impossible to make twelve out of any number that is made up of a number other than 2 or 3.

So, due to the prime factors of ten and twelve, ten can be made up of 1, 2, 5, and ten(2*5). While twelve can be made up of 1, 2, 3, 4(2*2), 6(2*3), and twelve(2*2*3).

  • Note: A common way to represent ten is X, especially when counting in base twelve, and as 10 in base twelve represents twelve, X0 would mean “ten twelves”. although I pronounce it Exty, or Desty, or Tenty, or whatever you want really. E is typically used to represent eleven, and pronounced “El” or “Elv”. I also like using R when referring to base twelve, because 12 is a number that’s dependant on the base it’s in, while R is clearly one stroke, and also looks kind of like 12 🙂

We could also find those prime factors by dividing the number successively, by each prime number, until we can no longer divide anymore. When doing this, we always start with the smallest prime, if only because it’s easier 🙂 So the first step is to divide any even number by 2 until it’s no longer even, then by 3 until it can no longer be divisible by 3, then 5, 7, etcetera. Once we get to one last prime number, we’ve found all the numbers prime factors, and from there we know everything there is to know about the factors for that number.

However, that’s not the only way we can find all the prime factors for a number.

We can also multiply numbers together.

Since we know that twelve is 2*2*3; and we know that ten is 2*5, if we wanted, we could multiply twelve and ten to get 2*2*2*3*5, a number which in base ten(X) is 120, and in base twelve(R) is X0.

Now, without doing anything else, we already know the prime factors for this number. They are three 2s, one 3, and one 5. Because those are the prime factors of the number we made it from. If we wanted to, we could find every non-prime number that can be used to make it up, by putting the prime factors along a square, multiplying them together as we go down and across so we have every prime, and every combination of primes found on one axis.

 

120 X0
Base  X Base R
1 2 4 8 1 2 4 8
3 6 12 24 3 6 10 20
5 12 24 40 5 X 18 34
15 30 60 120 13 26 50 X0

And now we know absolutely every possible factor of the number 120(X) X0(R), in both base X and base R. Or in any base greater than 5, we could just use 2*2*2*3*5, and that’s also guaranteed to represent the same number. The prime factors are the smallest possible unit you can possibly break a number into. So knowing how to represent the prime numbers in a base you’re working in is rather helpful. Note that prime numbers are prime regardless of base. It’s a fundamental property of the number itself, not how it’s represented.

53(X) and 45(R) both represent the same prime number, even though they look rather different, and neither representation is prime in the other base.

In base X, 45 is the composition of 3*3*5. Similarly, in base R, 53 is the composition of 3*3*7.

Actually, in base R, anything ending in 3, 6, or 9 or 0 has 3 as a prime factor, just like anything ending in 5 or 0 has 5 has a prime factor in base X.

And in either base, 2, 4, 6, 8, (X) and 0 are always even, and therefore can be divided by 2 before anything else.

Now, it’s most likely not obvious at first, but there’s something particularly special about base R when it comes to discovering primes. Here’s a chart of every prime up to 105(R) 149(X)

1 2 3 5 7 E 11 15 17 1E 25 27 31 35 37 3E 45 4E 51 57 5E 61 67 6E 75 81 85 91 95 X7 XE E5 E7 105
1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 109 113 127 131 137 139 149

Now here’s the first 100(R) numbers, with arranged by twelves, with the primes highlighted.

 Base X 1  Base R 1
2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 X E 10 11
14 15 16 17 18 19 20 21 22 23 24 25 12 13 14 15 16 17 18 19 1X 1E 20 21
26 27 28 29 30 31 32 33 34 35 36 37 22 23 24 25 26 27 28 29 2X 2E 30 31
38 39 40 41 42 43 44 45 46 47 48 49 32 33 34 35 36 37 38 39 3X 3E 40 41
50 51 52 53 54 55 56 57 58 59 60 61 42 43 44 45 46 47 48 49 4X 4E 50 51
62 63 64 65 66 67 68 69 70 71 72 73 52 53 54 55 56 57 58 59 5X 5E 60 61
74 75 76 77 78 79 80 81 82 83 84 85 62 63 64 65 66 67 68 69 6X 6E 70 71
86 87 88 89 90 91 92 93 94 95 96 97 72 73 74 75 76 77 78 79 7X 7E 80 81
98 99 100 101 102 103 104 105 106 107 108 109 82 83 84 85 86 87 88 89 8X 8E 90 91
110 111 112 113 114 115 116 117 118 119 120 121 92 93 94 95 96 97 98 99 9X 9E X0 X1
122 123 124 125 126 127 128 129 130 131 132 133 X2 X3 X4 X5 X6 X7 X8 X9 XX XE B0 E1
134 135 136 137 138 139 140 141 142 143 144 E2 E3 E4 E5 E6 E7 E8 E9 EX EE 100 101

If we wanted to make it even more obvious we could put it in base 6, then there’d just be two columns of primes, one on either side of the 10.

The reason for this is that compositions including 2 and 3 make up 2/3 of all numbers, and  they synchronize to allow only a specific 1/3 of numbers to get around them..

?? 2 3 2*2  ??? 2*3 ??? 2*2*2 3*3 2  ??? 2*2*3 ??? 2

This pattern repeats indefinitely, forcing any new prime number to have to first qualify to fit in one of those spots, then to be in a spot that isn’t already filled by another prime number.

So, another way we can build a number with a known composition of primes is to pick the primes first, then multiply them together 🙂

What primes a number has are important, especially when working with a base. Numbers’ sharing them makes a number easy to work with. That’s why fives and tens are so convenient when dealing in base ten.

In base twelve, twos, threes, fours, and sixes, get the same usefulness as fives. That’s because they’re all made of 2s and 3s in a ratio smaller than is in twelve, and so they all loop around every time we get to 10, just like 2 and 5 do in base ten.

So, we know we can tear apart a number to figure out what primes they’re made of.

We know we can build a number directly from primes.

We know we can take smaller numbers with known primes and multiply them together, which lets us keep all the primes of both numbers

Something else we can do, is take a single number with known primes, and multiply it by itself, that is, raising the number to an exponent, or squaring it(if the exponent is two and we only do it once)

For instance, 10 times 10 is 100. 10 times 10 is always 100, no matter what base we’re working with, it’s a fundamental property of our style of writing digits to represent things. Anything else writing it can get kind of tricky 🙂 But let’s work with 10 for now, and technically what we learn can be applied to ANY number we choose to square,

Let’s say we’re working in the old decimal base, 10 times 10 is therefore 84(X*X in R), but written as 100. The factors of 10(X), are 2 and 5. No matter what base we’re working with, if we can write 2 and 5, our factors are 2 and 5.

So what happened when we multiplied 10 by 10 to get 100? We added a 0. We took 10^2, and most importantly, we multiplied 2*5 by 2*5. That’s kind of a fundamental property of multiplication, as long as you’re still multiplying/diving, you can split numbers up as much as you darn well please, and toss them in any order, and still get the same answer.

That 2*5 * 2*5  could also be written as 2*2*5*5, to keep the numbers in one group. Another way it can be written is 2^2 * 5^2. These all = 100 in base X, and 84 in base R. Coincidentally, they also equal 10^2 and X^2.

So a number squared, is equal to each of its factors squared. Actually, a number to any power is equal to all of its prime factors taken to the same power. And that means we know how many numbers we can divide any power of ten by. Any number of 5s or 2s, as long as there aren’t more of them than we have 0s, and don’t include any other prime number.

So let’s make another table, six twos in the top row and 6 fives in the left column.

1 2 4 8 16 32 64
5 10 20 40 80 160 320
25 50 100 200 400 800 1600
125 250 500 1000 2000 4000 8000
625 1250 2500 5000 10000 20000 40000
3125 6250 12500 25000 50000 100000 200000
15625 31250 62500 125000 250000 500000 1000000

And there we have it. Every single number that can go evenly into a factor of ten up to the sixth power, with a million containing a total of 47(X) evenly divisible numbers(not including 1 and itself)

Tables are fun, let’s make another. This time we’ll use base twelve. From what we learned earlier, 2^6 * 2^6 * 3^6 should give us every prime in 1, 000,000 or 10  2s(12 2s in X) and six 6s

1 2 4 8 14 28 54 A8 194 368 714 1228 2454
3 6 10 20 40 80 140 280 540 X83 1940 3680 7140
9 16 30 60 100 200 400 800 1400 2800 5400 X800 19400
23 46 90 160 300 600 1000 2000 4000 8000 14000 28000 54000
69 116 230 460 900 1600 3000 6000 10000 20000 40000 80000 140000
183 346 690 1160 2300 4600 9000 16000 30000 60000 100000 200000 400000
509 X16 1830 3460 6900 11600 23000 46000 90000 160000 300000 600000 1000000

And there’s every factor of a dozenal million in base twelve. A total of 89(X) evenly divisible numbers to a dozenal million, not counting 1 and itself.  The top row is two times itself twelve times, or 2^10 (2^12 in base X: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2056, 4096), and the left column is every power of 3 up to 3^6(3, 9, 27, 81, 243, 729 in base X)

February 24th, 2014 | Tags: , , , .
Categories: Math | Comments: No Comments |