commit: c27249132ebb72b78a6ad939b3f928bc09409e8e
parent: cc8f13ec27ee72c599e072d9a28445c4e9ef263d
author: Chris Noxz <chris@noxz.tech>
date: Thu, 7 Dec 2023 19:01:28 +0100
add note about patterns in divisibility algorithms
1 file changed, 50 insertions(+), 2 deletions(-)
diff --git a/noxz.tech/articles/divisibility_theorems_and_fraction_flipping/index.www b/noxz.tech/articles/divisibility_theorems_and_fraction_flipping/index.www
@@ -1154,7 +1154,7 @@ algorithm with divisibility by 27 using
.I n
= -8. The next working number after 27 is, in fact, 29, and all prime numbers
appear to be included when I tested numbers up to 100. However, as I mentioned,
-I have not algebraically proven them.
+I have not algebraically proven them. There are in fact some other patterns...
.HnS 2
.TAG "math-rev-div-alg-script"
@@ -1178,7 +1178,7 @@ for div in range(1, 100):
num = div * 123456789123456
# perform recursive calculations while number is divisible by the div
- while num % div == 0 and len(str(num)) > 1:
+ while num % div == 0 and num > div:
# keep track of the previous number
prev = num
@@ -1206,6 +1206,54 @@ for div in range(1, 100):
.COE
.CDE
+When running the script, you might discover some interesting patterns from 11
+and up:
+
+.CDS
+.COS
+ 1 ---> 11: -1 |-
+ 3 | 13: +4 <---
+ +10 7 | 17: -5 <-- |
+ 9 | 19: +2 |+ | | +3
+ 1 ---> 21: -2 |- | -3 |
+ 3 | 23: +7 | <---
+ +10 7 | 27: -8 <-- |
+ 9 | 29: +3 |+ | | +3
+ 1 ---> 31: -3 |- | -3 |
+ 3 | 33: +10 | <---
+ +10 7 | 37: -11 <-- |
+ 9 | 39: +4 |+ | | +3
+ 1 ---> 41: -4 |- | -3 |
+ 3 | 43: +13 | <---
+ +10 7 | 47: -14 <-- |
+ 9 | 49: +5 |+ | | +3
+ 1 ---> 51: -5 |- | -3 |
+ 3 | 53: +16 | <---
+ +10 7 | 57: -17 <-- |
+ 9 | 59: +6 |+ | | +3
+ 1 ---> 61: -6 |- | -3 |
+ 3 | 63: +19 | <---
+ +10 7 | 67: -20 <-- |
+ 9 | 69: +7 |+ | | +3
+ 1 ---> 71: -7 |- | -3 |
+ 3 | 73: +22 | <---
+ +10 7 | 77: -23 <-- |
+ 9 | 79: +8 |+ | | +3
+ 1 ---> 81: -8 |- | -3 |
+ 3 | 83: +25 | <---
+ +10 7 | 87: -26 <-- |
+ 9 | 89: +9 |+ | | +3
+ 1 ---> 91: -9 |- | -3 |
+ 3 | 93: +28 | <---
+ 7 | 97: -29 <--
+ 9 | 99: +10 |+
+.COE
+.CDE
+
+Even if the code won't include 3 and 9, they do follow the same pattern.
+And the same could technically be said about 1. So, the pattern actually starts
+at 1!
+
Happy hunting!
.HnS 2
