“Number is the ruler of forms and ideas, and the cause of gods and dæmons.”
—Attributed to Pythagoras, The Sacred Discourse (born c.570 BC, died c.495 BC) by Iamblichus, in The Life of Pythagoras (c.300), translated by Thomas Taylor (1818)

Numbers are conventionally arrayed on a line, from 0 heading up towards an arbitrary amount that we term infinity. This format is limiting and only displays clearly additive relationships between numbers. As an alternative way of viewing numbers, which emphasises instead the multiplicative relationships between numbers, I propose a multi-dimensional space.

n-Dimensional Space

Start with a single axis, starting at the origin and stretching out to the right. The origin is 1. The first position is 2; the second position is 4; the third position is 8; and so on. This line is the power of 2 axis, that is, it is a log-2 scale. (The line can also stretch to the left, where the first leftward position is ½ or 0.5, the second leftward position is ¼ or 0.25, and so on.) Unlike in a normal number line, only positive numbers are expressed on this line, but with that caveat, all positive numbers are expressed on the line. Not all integers are at regular points, but all regular points above the origin are integers.

Next, we will add a second axis, stretching upwards. This axis will start at the same origin, 1, but the first position is 3, the second position is 9, the third position is 27, and so on. This line is the power of 3 axis, that is, it is a log-3 scale. (Again, the line can also stretch to the left, where the first leftward position is ⅓ or 0.33…, and so on.)

Numbers are now expressed by lines on the graph, rather than points. The number 1 is expressed by a line going through the origin: for example, when $x=1$, $y={\log_3 0.5}=-0.631…$. Again, all regular points above the origin are integers. The simplest regular point not on an axis is the number 6. $6=2\times3$, and is therefore at the point {1,1} on the graph (where the first axis, i.e. 2, is listed first). Its line crosses the 2-axis at 2.585…, i.e. $\log_2 6$, and crosses the 3-axis at 1.631…, i.e. $\log_3 6$.

We can continue adding dimensions indefinitely. Add a third axis, for 4; you can imagine it stretching backwards. Now each number is expressed by a plane, not a line. The plane for 4 now has two non-negative integer solutions: {2,0,0} and {0,0,1}, that is, (2²)*(3⁰)*(4⁰)=4*1*1=4 and (2⁰)*(3⁰)*(4¹)=1*1*4=4. When the number of the plane is less than or equal to the number of axes (plus one, for unity), the number of non-negative integer solutions is the number of ways that a number's factors can be multiplied together to produce the number. So for the 12-plane on the 11-dimensional space (i.e. 2-axis to 12-axis) there are the following solutions:

{0,0,0,0,0,0,0,0,0,0,1}=(2⁰)*(3⁰)*(4⁰)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12¹)=1*1*1*1*1*1*1*1*1*1*12=12
{1,0,0,0,1,0,0,0,0,0,0}=(2¹)*(3⁰)*(4⁰)*(5⁰)*(6¹)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=2*1*1*1*6*1*1*1*1*1*1=12
{0,1,1,0,0,0,0,0,0,0,0}=(2⁰)*(3¹)*(4¹)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=1*3*4*1*1*1*1*1*1*1*1=12
{2,1,0,0,0,0,0,0,0,0,0}=(2²)*(3¹)*(4⁰)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=4*3*1*1*1*1*1*1*1*1*1=12

So this is n-dimensional space (or n-space for short): a way to visualise numbers graphically in a way that emphasises their multiplicative (and especially power) relations. I find it an interesting way to reconceptualise numbers.

p-Dimensional Space

But there is a special subset of n-space that is even more interesting and useful. In the n=12 example above, there were four non-negative integer solutions—but (of course) only one of those involved exclusively prime factors. Therefore if we remove all of the non-prime axes, we will find that every positive integer has a unique non-negative integer solution in this truncated n-space. I call this truncated n-space, “prime-dimensional space”, (or p-dimensional space, or p-space for short). Here are the first solutions for the numbers. The solutions are expressed up to their maximum positive axis, that is, all unstated values after the last listed are to be understood as being 0.

={0}
={1}
={0,1}
={2}
={0,0,1}
={1,1}
={0,0,0,1}
={3}
={0,2}
={1,0,1}
={0,0,0,0,1}
={2,1}
={0,0,0,0,0,1}
={1,0,0,1}
={0,1,1,0}
={4}
={0,0,0,0,0,0,1}
={1,2}
={0,0,0,0,0,0,0,1}
={2,0,1}

This is an even more elegant way of visualising numbers multiplicatively. The multiples nestle compactly on the first few axes, while the primes stretch out into increasingly isolated axes. With each number having a unique solution, we can count up through the numbers, pointing to their positions as we go.

Each p-dimensional plane for $n$ can be described by an equation as follows:

$\sum_{i = 2}^{p} \frac{x_{i}}{\log_{i} (n)} = 1$

Prime Numbers

If we only plot each number on a grid up to the n/2 axis (for example, plot 12 up to the 5-axis, as it is the last prime before ${12\over2}=6$, or 14 up to the 7 axis, as ${14\over2}=7$), then only multiples have a Diophantine solution. So, for example, 6 is a multiple because there is an integer solution ($x=1$, $y=1$) to this equation:

$\frac{x}{\log_{2} (6)} + \frac{y}{\log_{3} (6)} = 1$

But 7 is prime because there is no integer solution to this equation:

$\frac{x}{\log_{2} (7)} + \frac{y}{\log_{3} (7)} = 1$

This is, to my knowledge, an entirely novel way of looking at the question of prime numbers. If it is possible to know whether there are solutions to this group of equations, then we can always factorise multiples and identify prime numbers, and the inverse applies. This allows us to use the tools and techniques of Diophantine analysis to attack the problem of primes.

sum θoətiz abaʊt praim-daimenʃənəl speis

“numbə bii ðə rʊʊlər ov fʊəmiz and aidiəriz, and ðə kʊəz ov godiz and diiməniz.”
—atribyʊʊtəð tə paiθagərəs, ðə seikrəd diskʊəs (buəθəð s.570 b.k., daiyəð s.495 b.k.) baiy iiyamblikəs, in ðə laif ov paiθagərəs (s.300), transleitəð bai toməs teilə (1918)

numbəriz bii kənvenʃənəliiy areiyəð on a lain, from 0 hediŋg up təwʊədz an aəbitreriiy amaʊnt ðat wii tuəm infinitii. ðis fʊəmat bii limitiŋg and oʊnlii displei kliəliiy aditiv rileiʃənʃipiz bitwiin numbəriz. az an oltuənətiv weiy ov vyʊʊwiŋ numbəriz, wic emfəsaiz insted ðə multiplikətiv rileiʃənʃipiz bitwiin numbəriz, mii prəpoʊz a multii-daimenʃənəl speis.

n-daimenʃənəl speis

staət wið a singəl aksis, staətiŋg at ðiiy orijin and streciŋg aʊt tə ðə rait. ðiiy orijin biiy 1. ðiiy umθ pəziʃən bii 2; ðə tʊʊθ pəziʃən bii 4; ðə tiiθ pəziʃən bii 8; and soʊw on. ðis lain bii ðə paʊwər ov 2 aksis, ðat bii, hii biiy a log-2 skeil. (ðə lain kan oəlsoʊ strec tə ðə left, weə ðiiy umθ leftwəd pəziʃən biiy ½ oə 0.5, ðə tʊʊθ leftwəd pəziʃən bii ¼ oə 0.25, and soʊw on.) unlaik in a noəməl numbə lain, oʊnlii pozitiv numbəriz biiy ekspresəð on ðis lain, but wið ðat kaviiyat, oəl pozitiv numbəriz biiy ekspresəð on ðə lain. not oəl intəjəriz biiy at regyələ pointiz, but oəl regyələ pointiz abuv ðiiy orijin biiy intəjəriz.

nekst, wi adil a tʊʊθ aksis, streciŋg upwədz. ðis aksis staətil at ðə seim orijin, 1, but ðiiy umθ pəziʃən bii 3, ðə tʊʊθ pəziʃən bii 9, ðə tiiθ pəziʃən bii 27, and soʊw on. ðis lain bii ðə paʊwər ov 3 aksis, ðat bii, hii biiy a log-3 skeil. (agen, ðə lain kan oəlsoʊ strec tə ðə left, weə ðiiy umθ leftwəd pəziʃən biiy ⅓ oə 0.33…, and soʊw on.)

numbəriz bii naʊw ekspresəð bai lainiz on ðə graf, raəðə ðan pointiz. ðə numbər 1 biiy ekspresəð baiy a lain goʊwiŋ θrʊʊ ðiiy orijin: foər egzampəl, wen $x=1$, $y={\log_3 0.5}=-0.631…$. agen, oəl regyələ pointiz abuv ðiiy orijin biiy intəjəriz. ðə simplist regyələ point not on an aksis bii ðə numbə 6. $6=2\times3$, and bii ðeəfʊə at ðə point {1,1} on ðə graf (weə ðiiy umθ aksis, i.e. 2, bii listəð umθ). hiis lain kros ðə 2-aksis at 2.585…, i.e. $\log_2 6$, and kros ðə 3-aksis at 1.631…, i.e. $\log_3 6$.

wii kan kəntinyʊʊw adiŋ daimenʃəniz indefinətliiy. ad a tiiθ aksis, foə 4; ðii kan imajən hii streciŋ bakwədz. naʊw iic numbə biiy ekspresəð baiy a plein, not a lain. ðə plein foə 4 naʊ hav tʊʊ non-negətiv intəjə səlʊʊʃəniz: {2,0,0} and {0,0,1}, ðat bii, (2²)*(3⁰)*(4⁰)=4*1*1=4 and (2⁰)*(3⁰)*(4¹)=1*1*4=4. wen ðə numbər ov ðə plein bii les ðan oər iikwəl tə ðə numbər ov aksisiz (plus um, foə yʊʊnitii), ðə numbər ov non-negətiv intəjə səlʊʊʃəniz bii ðə numbər ov weiyiz ðat a numbər-iis faktəriz kan bii multiplaiyəð təgeðə tə prəjʊʊs ðə numbə. soʊ foə ðə 12-plein on ðə 11-daimenʃənəl speis (i.e. 2-aksis tə 12-aksis) ðeə bii ðə foloʊwiŋ səlʊʊʃəniz:

{0,0,0,0,0,0,0,0,0,0,1}=(2⁰)*(3⁰)*(4⁰)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12¹)=1*1*1*1*1*1*1*1*1*1*12=12
{1,0,0,0,1,0,0,0,0,0,0}=(2¹)*(3⁰)*(4⁰)*(5⁰)*(6¹)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=2*1*1*1*6*1*1*1*1*1*1=12
{0,1,1,0,0,0,0,0,0,0,0}=(2⁰)*(3¹)*(4¹)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=1*3*4*1*1*1*1*1*1*1*1=12
{2,1,0,0,0,0,0,0,0,0,0}=(2²)*(3¹)*(4⁰)*(5⁰)*(6⁰)*(7⁰)*(8⁰)*(9⁰)*(10⁰)*(11⁰)*(12⁰)=4*3*1*1*1*1*1*1*1*1*1=12

soʊ ðis bii n-daimenʃənəl speis (oə n-speis foə ʃʊət): a wei tə viʒyʊʊwəlaiz numbəriz grafikliiy in a wei ðat emfasaiz diis multiplikətiv (and espeʃəlii paʊwə) rileiʃəniz. mii faind hiiy an intərestiŋg wei tə riikənaepcʊʊwəlaiz numbəriz.

p-daimenʃənəl speis

but ðeə biiy a speʃəl subset ov n-speis ðat biiy iivən mʊər intərestiŋg and yʊʊsfəl. in ðə n=12 egzampəl abuv, ðeə biiyid fʊ non-negətiv intəjə səlʊʊʃəniz—but (ov cʊəs) oʊnliiy um ov ðoʊʒ involvid eksklʊʊsivlii praim faktəriz. ðeəfoər if wii rimʊʊv oəl ov ðə non-praim aksisiz, wi faindil ðat evrii pozitiv intəjə hav a yʊʊniik non-negətiv intəjə səlʊʊʃən in ðis trunkeitəð n-speis. mii koəl ðis trunkeitəð n-speis, “praim-daimenʃənəl speis”, (oə p-daimenʃənəl speis, oə p-speis foə ʃʊət). hiə bii ðiiy umθ səlʊʊʃəniz foə ðə numbəriz. ðə səlʊʊʃəniz biiy ekspresəð up tə diis maksiməm pozitiv aksis, ðat biiy, oəl unsteitəð valyʊʊwiz aftə ðə last listəð bii tə biiy undəstandəð az biiyiŋ 0.

={0}
={1}
={0,1}
={2}
={0,0,1}
={1,1}
={0,0,0,1}
={3}
={0,2}
={1,0,1}
={0,0,0,0,1}
={2,1}
={0,0,0,0,0,1}
={1,0,0,1}
={0,1,1,0}
={4}
={0,0,0,0,0,0,1}
={1,2}
={0,0,0,0,0,0,0,1}
={2,0,1}

ðis biiy an iivən mʊər eləgənt weiy ov viʒʊʊwəlaiziŋ numbəriz multiplikətivlii. ðə multipəliz nesəl kompaktliiy on ðiiy umθ fyʊʊw aksisiz, wail ðə praimiz strec aʊt intʊʊw inkriisiŋgliiy aisəleitəð aksisiz. wið iic numbə haviŋg a yʊʊniik səlʊʊʃən, wii kan kaʊnt up θrʊʊ ðə numbəriz, pointiŋ tə diis pəziʃəniz az wii goʊw.

iic p-daimenʃənəl plein foə $n$ kan bii diskraibəð baiy an iqweiʒən az foloʊ:

$\sum_{i = 2}^{p} \frac{x_{i}}{\log_{i} (n)} = 1$

praim numbəriz

if wiiy oʊnlii plot iic numbər on a grid up tə ðə n/2 aksis (foər egzampəl, plot 12 up tə ðə 5-aksis, az hii bii ðə last praim bifʊə ${12\over2}=6$, oə 14 up tə ðə 7 aksis, az ${14\over2}=7$), ðen oʊnlii multipəliz hav a daiyəfantiin səlʊʊʃən. soʊ, foər egzampəl, 6 biiy a multipəl bikuz ðeə biiy an intəjə səlʊʊʃən ($x=1$, $y=1$) tə ðis ikweiʒən:

$\frac{x}{\log_{2} (6)} + \frac{y}{\log_{3} (6)} = 1$

but 7 bii praim bikuz ðeə bii noʊw intəjə səlʊʊʃən tə ðis ikweiʒən:

$\frac{x}{\log_{2} (7)} + \frac{y}{\log_{3} (7)} = 1$

ðis bii, tə miis nolij, an entaiəli novəl weiy ov lʊkiŋg at ðə kwescən ov praim numbəriz. if ðis bii posibəl tə noʊ weðə ðeə bii səlʊʊʃəniz tə ðis grʊʊp ov ikweiʒəniz, ðen wii kan oəlweiz faktəraiz multipəliz and aidentifai praim numbəriz, and ðiiy invuəs aplai. ðis alaʊ wii tə yʊʊz ðə tʊʊliz and tekniikiz ov daiyəfantiin analisis tʊʊw atak ðə probləm ov praimiz.