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Factorial Quotient (Posted on 2011-11-19) Difficulty: 3 of 5
Find a positive integer p such that: (p+1)(p+2)....(p+500)/500! is an integer with no prime factors less than 500.

No Solution Yet Submitted by K Sengupta    
Rating: 4.5000 (2 votes)

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re: Better solution? (spoiler) Comment 4 of 4 |
(In reply to Better solution? (spoiler) by Steve Herman)

You're absolutely correct that my method doesn't return the minimal value for p for each N. In fact, I am most certain that there is a smaller value than my mammoth solution for N = 500. However, I like you, was just trying to find a way to make sense of it and get an answer. Looking at the computed minimal values for p for small values of n, I can't seem to find any logical pattern in the results, but I'll present them here if you wish to pursue it.


Perhaps, at this time, a switch to C++ from Python would be a good idea, as computation times will be much improved. Although, it would help if I smartened up my routines a little, and didn't factor every number every time (ie 3*4*5*6/4! ... then factor again for 4*5*6*7/4!).

Anyway, here's the minimal p values for 2 < n < 43:

n = 3; p = 4
n = 4; p = 3
n = 5; p = 18
n = 6; p = 56
n = 7; p = 136
n = 8; p = 36
n = 9; p = 150
n = 10; p = 36
n = 11; p = 36
n = 12; p = 162
n = 13; p = 2226
n = 14; p = 225
n = 15; p = 704
n = 16; p = 225
n = 17; p = 5832
n = 18; p = 2080
n = 19; p = 2080
n = 20; p = 43176
n = 21; p = 14850
n = 22; p = 19552
n = 23; p = 35400
n = 24; p = 193025
n = 25; p = 2080
n = 26; p = 36261
n = 27; p = 1092
n = 28; p = 256
n = 29; p = 240450
n = 30; p = 58752
n = 31; p = 341056
n = 32; p = 371910
n = 33; p = 6426
n = 34; p = 69580
n = 35; p = 37584
n = 36; p = 152152
n = 37; p = 152152
n = 38; p = 487305
n = 39; p = 767880
n = 40; p = 85701
n = 41; p = 3017280
n = 42; p = 96580

EDIT:
n = 43; p = 24041556
n = 44; p = 45043155
n = 45; p = 9484050
n = 46; p = 692176
n = 47; p = 232906752
n = 48; p = 45375176
n = 49; p = 38074050
n = 50; p = 4302156
n = 51; p = 13927628
n = 52; p = 366795
n = 53; p = 79221186
n = 54; p = 7638400
n = 55; p = 53583040
n = 56; p = 17868930
n = 57; p = 34296386
n = 58; p = 4703041
n = 59; p = 108178500
n = 60; p = 93851136
n = 61; p = 2237874562
n = 62; p = 254322432
n = 63; p = 157776256
n = 64; p = 266194435
n = 65; p = 174133806
n = 66; p = 25013376
n = 67; p = 673750800

Edited on November 21, 2011, 6:10 pm
  Posted by Justin on 2011-11-20 20:19:17

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