## Integer Factorization: The Jump Chart for the Trip to 111000011

… As a tool, to help me figure out how to do it in ONE JUMP… i.e., a formula for the general case so I can look at the Ceiling Root, along with the first few remainders (or just one of them if it’s a really good formula!), and come up with the magic number 268458 to add to the Ceiling Root.

This method of looking at the first remainder and every second one after that fits the metaphor, as I commented on Facebook, of lightspeed-skipping. It’s a good way to set your Corellian cruiser on fire. 😀 I hope to do better.

``````m = 111000011.
Ascent will be 268458 = 2*3*101*443, requiring half that many jump-trips. This is the number
I want to calculate.
First 51 remainders (the values of x^2+21072x+7285 for x=0 to 50):
7285 28358 49433 70510 91589 112670 133753 154838 175925 197014 218105 239198 260293 281390 302489 323590 344693 365798 386905 408014 429125 450238 471353 492470 513589 534710 555833 576958 598085 619214 640345 661478 682613 703750 724889 746030 767173 788318 809465 830614 851765 872918 894073 915230 936389 957550 978713 999878 1021045 1042214 1063385

Jump 1: Start at 7285. Go 111 to 7396. Jump 221^2-86^2 = 41445 to 48841. Go 592.
Sub-jump trips are 3*37 and 2^4*37.
# Perfect Squares passed: 136 = 2^3*17.  Intersquare avg. 307.
Jump 2: Start at 49433. Go 296 to 49729. Jump 302^2-222^2 = 41920 to 91204. Go 385.
Sub-jump trips are 8*37 and 5*7*11. Difference from last jump is 5^2*19.
# Perfect Squares passed: 81 = 3^4. Intersquare avg. 524.
Jump 3: Start at 91589. Go 220 to 91809. Jump 365^2-303^2 = 41416 to 133225. Go 528.
Sub-jump trips are 2^2*5*11 and 2^4*3*11. Difference from last jump is -(2^3*3^2*7).
# Perfect Squares passed: 63 = 3^2*7. Intersquare avg. 668.
Jump 4: Start at 133753. Go 203 to 133956. Jump 418^2-366^2 = 40768 to 174724. Go 1201.
Sub-jump trips are 7*29 and 1201. Difference from last jump is -(2^3*3^4).
# Perfect Squares passed: 53. Intersquare avg. 784.
Jump 5: Start at 175925. Go 475 to 176400. Jump 467^2-420^2 = 41689 to 218089. Go 16.
Sub-jump trips are 5^2*19 and 2^4. Difference from last jump is 3*307.
# Perfect Squares passed: 48 = 2^4*3. Intersquare avg. 887.
Jump 6: Start at 218105. Go 919 to 219024. Jump 510^2-468^2 = 41076 to 260100. Go 193.
Sub-jump trips are 919 and 193. Difference from last jump is 613. (All prime!)
# Perfect Squares passed: 43. Intersquare avg. 978.
Jump 7: Start at 260293. Go 828 to 261121. Jump 549^2-511^2 = 40280 to 301401. Go 1088.
Sub-jump trips are 2^2*3^2*23 and 2^6*17. Difference from last jump is -(2^2*199).
# Perfect Squares passed: 39. Intersquare avg. 1060.
Jump 8: Start at 302489. Go 11 to 302500. Jump 587^2-550^2 = 42069 to 344569. Go 124.
Sub-jump trips are 11 and 2^2*31. Difference from last jump is 1789.
# Perfect Squares passed: 38. Intersquare avg. 1137.
Jump 9: Start at 344693. Go 1051 to 345744. Jump 622^2-588^2 = 41140 to 386884. Go 21.
Sub-jump trips are 1051 and 3*7. Difference from last jump is -929.
# Perfect Squares passed: 35. Intersquare avg. 1210.
Jump 10: Start at 386905. Go 1224 to 388129. Jump 655^2-623^2 = 40896 to 429025. Go 100.
Sub-jump trips are 2^3*3^2*17 and 2^2*5^2. Difference from last jump is -(2^2*61).
# Perfect Squares passed: 33. Intersquare avg: 1278.
``````