It is often easier to disprove a conjecture than to try and prove it to be true. You have but to find the exception to the conjecture. We received only one contribution disproving the conjecture and that was the work of Tim from Gravesend Grammar School for Boys in Gravesend, Kent. His work is quite detailed and is as follows:

A computer search showed that this is not true: the first example is 1002, which is 167*6, has eight factors (1, 2, 3, 6, 167, 334, 501, 1002) and 1001 also has eight factors (1, 7, 11, 13, 77, 91, 143, 1001). The next example is 1086 and 1085, and after that 1266 and 1275. The first example where an adjacent number has more factors than the multiple of six is 2274, which has 8 factors, and 2275, which has 12 factors. The number 6546 has 8 factors, while 6545 has 16. This is the computer program used to find these numbers, written in Microsoft Visual Basic (a list box called list 1 needs to be created on the form, the factors are then displayed in this list box):

 n = 1
1 f = 0
fa = 0
fb = 0
for a = 1 to 6*n
If Int(6*n/a) =  6*n/a Then f = f + 1: list1.Addltem a
Next
For a = 1 To 6*n-1
If Int((6*n - 1)/a) = (6*n-1)/a then fa = fa + 1: list1.Addltem a
Next
For a = 1 To 6*n+1
If Int((6*n + 1)/a) = (6*n + 1)/a Then fb = fb + 1: list1.Addltem a
Next
DoEvents
If fa > f + 4 Or fb > f+4 Then MsgBox "YES": GoTo 2
3 n = n + 1
list 1.Clear
GoTo 1
2 form1.show
DoEvents
MsgBox " "
GoTo 3



Well done, Tim. However, for those of you that may not know how to program in Microsoft Visual Basic, you might like to use the following BBC BASIC program:

110 INPUT N
120 FOR F = 1 TO N/2
130 IF INT(N/F) = N/F THEN PRINT F
140 NEXT F
150 GOTO 110


This program is not as nice as Tim's but it will give you the factors of a number which is N in the program. Then if you consider that you are after a number which is either one more than or one less than a multiple of 6, choose a multiple of 6, such a 6*p, where p is a large prime number, and look at numbers on either side of this multiple. Choosing a prime multiple of 6 keeps the number of factors of the multiple as small as possible and your search is not so arduous.