Prime Number Folio Coordinate System
Updated: Jun 17
When I broke my brain, light entered through some of the cracks. My brain healed, but the light stayed. My past mental health struggles turned something on in my brain that made me smarter. You might want to review my other math posts to better understand how my mind works. You can find them here, here and here. In a word my brain is neurodivergent.
The attached image is from a Prime Number Folio Coordinate System that I created. Folio math is similar to modular math, but instead of the numbers wrapping around or spinning around a unit circle, they turn back at different positions on both the X and Y axis. In other words, they never make full cycles.
“The image stands on its own. The patterns should jump off the page. Especially with the color.”
The following are a list of observations about the Prime Number Folio Coordinate System. There’s an algorithm too. More than one. These observations are in no way mathematically rigorous. I may make a mistake or two in the description. I go off on a few tangents and make general observations. Even so, with the least bit of numeracy, you should be able to follow along. If you are a professionally trained mathematician, you’re going to have to be patient and try to follow how my mind processes this type of work. It’s not academic or professional, but it’s logical and true. If you notice a mistake, it’s just that, a mistake. It can be fixed.
The Y-Axis splits at the top, and the X-Axis splits on the left. The colors help this stand out.
Let’s start with the top of the Y-Axis.
All digits at the top of the Y-Axis reduce down to 1,7, 4 or 5,2, 8. This is important. Using this Prime Number Folio Coordinate System, it’s easier to think of prime numbers in separate sequences across from each other and right or left-handed rather than next to each other on a number line. I see them as Chiral.
All digits in on the right-hand side of the Y-Axis reduce down to 5, 2 or 8. (For example 179 has 3 digits, what matters is that the numbers 1 +7+9 sum to the number 8.) So this would be considered a right-handed prime number. Or a number on the right side of the Y-Axis.
Right-handed numbers have different properties than the left-handed numbers. For example…
The numbers on the right side (5,2,8) of the Y-Axis include not only prime numbers, but the products of the prime numbers combined from both sides of the Y-axis. Every product on the right-hand side of the Y-Axis is created from two primes (or semi-primes or combination of semi-primes) from both sides of the Y-axis (one from each side), which ALWAYS sum to an exact multiple of 6. These are plotted on the right side of the X-Axis. (For example 7 X 11 = 77. While 7+11= 18.) Using this Folio Coordinate System, it’s easy to see how the products and sums and their distribution are directly related to each other. You might want to start thinking about the Goldbach Conjecture. All products and sums on the right side are indigo/purple to show how they combine with the red and blue prime numbers.
The Prime Number Folio Coordinate System is a new number positioning system. This is a 2-Dimensional representation of the system. The easiest one to understand. It leaves the natural number line in the past where it belongs. It looks like we are simply adding 6 to each Axis/number line, when in fact we are adding the number 1 to each consecutive number but positioning it at different points while moving around both the X and Y Axis. The colors will help your eye follow the numbers. Follow the colors of the rainbow/number combination to help you move around the system.(R-1,O-2,Y-3,G-4,B-5,I-6)
The number 35 is an important number. It’s the first number on the right-hand side that’s a product of two prime factors of 5 X 7. The sum of 5+7 = 12. Since the right-handed numbers are distributed evenly by 6, we can add 42 (7X6) to 35 and land on the number 77. So now we know that starting with the number 35 if we add 42 continuously we will NEVER land on a prime number. We can also add 30(5×6) to 35 and land on 65. We also know that 5 + 13 =18 and 7+11= 18.
The next number that introduces a product of two primes is 65 (5X13) 5+13 = 18. So we can take 78(6X13) and add this to 65 and land on 143. Which is the product of (11X13) Starting with 65 we can add 78 continuously and NEVER land on a prime number.
In the meantime 77 (The product of 7 and 11 now introduces the prime number 11 into the mix. So 77 + 66 (6X11) = 143. Starting with 77 we can add 66 continuously and NEVER land on a prime number. **Note-You can’t add multiples of 6 until that multiple is introduced into the sequence. This may seem obvious to you, but it wasn’t to me.
While this is a great way to see where the primes will NEVER land, it’s not helpful in telling us where the primes will be distributed. But it actually is. Here’s how…
You need to frame the question like this. How many right-handed primes (5,2,8) are there up to 5X 49=245? (You can use 5 times any number from the left-handed side of the table) Counter-intuitively we use a multiple of 5 because we know that other than the number 5 no other prime number ends in 5. Moreover, no other prime number when multiplied by 6 ends in zero (6×5=30) Two unique and important features for this algorithm. Now all you have to do is continually subtract 30 (5X6) starting with 245 until you get all the way back down to 35. You have to go back to the beginning and then start adding/counting forward.
So now starting with 35 you add 42 until you land on the first number greater than 245, then STOP! At 77 you start adding 66 to the sequence until you reach the first number over 245, then STOP! At 65 you start adding 78 until you reach the first number over 245, then STOP! Once you reach the first multiple of 6 where the next added number is over 245 stop the sequence. The remaining numbers not landed on are the primes!! Simple. It’s a sorting algorithm. (NOTE*** SUBTRACTION ONLY WORKS WITH NUMBERS DIVISIBLE BY 5. OTHER THAN 5 IT’S IMPOSSIBLE TO TELL IF A NUMBER IS DIVISIBLE BY A PRIME JUST BY LOOKING AT IT, SINCE THEY ALL END IN 1,3,7,9. SO YOU HAVE TO ADD FOR THIS ALGORITHM TO WORK. IT ONLY WORKS ONE WAY, AND YOU HAVE TO KNOW WHERE TO START***) This statement is not entirely true, but it’s still helpful.
Keep in mind; this only includes the primes on the right-hand side of our Prime Number Folio Coordinate System. The primes on the left behave differently. You can still move around using multiples of 6, but there is no common starting point like the number 35. You have to start with the squares of 5 at 25(in blue)for one sequence of numbers and the square of 7 at 49(in red)for the other sequence of numbers. The sums of these products are also not exact multiples of 6. They sum to 10 and 14 and are matched to the split X Axis on the left-hand side of the graph. (See spreadsheet and the relationship between their products and sums. Keep thinking Goldbach Conjecture)
The left-handed primes (1,7,4) are also distributed regularly between multiples of 6 starting with the square number 25. All square number primes are distributed on the left-hand side of this table. The products are created from numbers from the same sides of the table. Either left or right.There are no crossover combinations. Hence the square numbers. A useful visual representation is to imagine making big pinwheel spins with your arms. For the right side of the table, they are spinning inwards, and your arms are crossing each other. For the left side of the table, they are spinning outward in separate circles, and the lines never intersect. They only touch at certain tangent points starting at 175 and then only with numbers having the common multiple of 5 and 7.
If you want to know if an exact number is prime, say 241. Just add by 6 till you reach the next multiple of 5. In this case 265(that sums to a 1,4,7) and use the same algorithm as on the right side. However, and this is a BIG; however, we also have to include the number 49 as a starting point when we add forward towards the number 241. You have two separate starting points for the left-handed sorting algorithm. There’s a reason no one has noticed this until now. It’s complicated and hidden in plain sight. These two algorithms combined should make finding prime numbers and a composite numbers prime factors as easy as pie. They just need to be programmed into a computer and tested for million digit numbers. It should still work, if it doesn’t, I fix it. No biggie.
All Odd numbers fall on the Y axis and All Even numbers on the X-axis. The Prime Number Folio Coordinate System and it’s natural numbers are all you need to find a prime number or a composite number and it’s factors. No need for complex numbers or the Reimann Hypothesis.
There are so many more mathematical treasures in this Prime Number Folio Coordinate System. You can probably see for yourself. Especially the even number distribution and growth on the X-Axis. Look how the number 2 raised to an odd number power all fall on the top of the X-Axis(orange) and how the number 2 raised to an even number power all fall on the bottom of the X-Axis(green.) The number 10 raised to odd and even powers all fall on the same bottom green line.
I haven’t even gotten started on the Y-3 line. Or how the squared prime numbers grow and are absolutely predictable, now that you know you have multiple starting points. Let’s not forget the cubes. Or the 6-I line. So much mathematical gold. This 2-Dimensional image only scratches the surface.
I can do things with numbers no one ever dreamed possible. All the numbers, even the ones no one knows exist. I see things you don’t. Email me for more info.