R0x0rk1tt3h Central

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)