** The Ulam Spiral **
Showing primes between 1 - 89872
Coil a string of whole numbers around itself:
17-----18-----19-----20-----21
16-----05-----06-----07-----22
15-----04-----01-----08-----23
14-----03-----02-----09-----24
13-----12-----11-----10-----25
Now look for perfect squares (1,4,9,16,25,36 ...) and Primes (Divisible only by 1 and N: N>1) The primes seem to form diagonal "strings". The squares form interesting spirals.
The above numeric spiral showing only primes.
17------------19--------------
-------05------------07-------
--------------01------------23
-------03-----02--------------
13------------11--------------
The Ulam spiral, named for Stanislaw Ulam, seems to show a large number of continuous diagonal 'strings' for prime numbers (2 divisors). The images below show the Ulam spiral for different number(s) of divisors.
In the below 17 images, each image shows whole numbers that start at 1 and end with 10002.
Divisors=2 (Prime) ; highest shown number with 2 divisors = 9973
Factors of 9973: 1,9973
Between 1 and 10002 || 1229 numbers have 2 divisors. (Prime)
Divisors=3 ; highest shown number with 3 divisors = 9409
Factors of 9409: 1,97,9409
Between 1 and 10002 || 25 numbers have 3 divisors.
Divisors=4 ; highest shown number with 4 divisors = 10001
Factors of 10001: 1,73,137,10001
Between 1 and 10002 || 2609 numbers have 4 divisors.
Divisors=5 ; highest shown number with 5 divisors = 2401
Factors of 2401: 1,7,49,343,2401
Between 1 and 10002 || 4 numbers have 5 divisors.
Divisors=6 ; highest shown number with 6 divisors = 9981
Factors of 9981: 1,3,9,1109,3327,9981
Between 1 and 10002 || 764 numbers have 6 divisors.
Divisors=7 ; highest shown number with 7 divisors = 729
Factors of 729: 1,3,9,27,81,243,729
Between 1 and 10002 || 2 numbers have 7 divisors.
Divisors=8 ; highest shown number with 8 divisors = 9994
Factors of 9994: 1,2,19,38,263,526,4997,9994
Between 1 and 10002 || 2114 numbers have 8 divisors.
Divisors=9 ; highest shown number with 9 divisors = 9025
Factors of 9025: 1,5,19,25,95,361,465,1805,9025
Between 1 and 10002 || 32 numbers have 9 divisors.
Divisors=10 ; highest shown number with 10 divisors = 9904
Factors of 9904: 1,2,4,8,16,619,1238,2476,4952,9904
Between 1 and 10002 || 150 numbers have 10 divisors.
Divisors=11 ; highest shown number with 11 divisors = 1024
Factors of 1024: 1,2,4,8,16,32,64,128,256,512,1024
Between 1 and 10002 || 1 number has 11 divisors.
Divisors=12 ; highest shown number with 12 divisors = 9999
Factors of 9999: 1,3,9,11,33,99,101,303,909,1111,3333,9999
Between 1 and 10002 || 1040 numbers have 12 divisors.
Divisors=13 ; highest shown number with 13 divisors = 4096
Factors of 4096: 1,2,4,8,16,32,64,128,256,512,1024,2048,4096
Between 1 and 10002 || 1 number has 13 divisors.
Divisors=14 ; highest shown number with 14 divisors = 9664
Factors of 9664: 1,2,4,8,16,32,64,151,302,604,1208,2416,4832,9664
Between 1 and 10002 || 41 numbers have 14 divisors.
Divisors=15 ; highest shown number with 15 divisors = 9801
Factors of 9801: 1,3,9,11,27,33,81,99,121,297,363,891,1089,3267,9801
Between 1 and 10002 || 15 numbers have 15 divisors.
Divisors=16 ; highest shown number with 16 divisors = 9982
Factors of 9982: 1,2,7,14,23,31,46,62,161,217,322,434,713,1426,4991,9982
Between 1 and 10002 || 800 numbers have 16 divisors.
Divisors=17
Between 1 and 10002 || No numbers have 17 divisors.
Divisors=18 ; highest shown number with 18 divisors = 9972
Factors of 9972: 1,2,3,4,6,9,12,18,36,277,554,831,1108,1662,2493,3324,4986,9972
Between 1 and 10002 || 159 numbers have 18 divisors.
Divisors=19
Between 1 and 10002 || No numbers have 19 divisors.
Divisors=20 ; highest shown number with 20 divisors = 9968
Factors of 9968: 1,2,4,7,8,14,16,28,56,89,112,178,356,623,712,1246,1424,2492,4984,9968
Between 1 and 10002 || 157 numbers have 20 divisors.
Start at 2500 (50^2) and show only squares (50^2, 51^2 . . . ) an interesting spiral appears when the start number is a square and all other plotted points along the Ulam spiral are squares.
Let us use an Archimedes spiral (r=theta| x=r cos theta| y=r sin theta) and plot intergers from 100 (Perfect Square) to 2963 (Prime).
| Blue=N+1 | | Red=Perfect Square | | Cyan=Prime |
So what is my point ?
I believe that this method of coiling whole numbers (The Ulam spiral) allows for a myriad of interesting patterns and has no significance to Prime numbers, it just happens. A lot has been written about the 'diagonal strings' that the Prime numbers form, look at number of divisors (3, 9, 15). This is an interesting display of 'diagonal strings'. It seems that all 'odd number divisors' show diagonal patterns. D=17 and D=19 yield no results between 1 and 10002. However, the Archimedes spiral shows a different 'prime' pattern that seems to display a better insight into the realm of primes!
Program information: The program(s) to generate the above images were written in GW Basic. A For-Next loop was used to define the number of divisors and if a point should be displayed. In general, with a Pentium III @ 700 MHz, it took 7 minutes to render each image.