Katedra matematiky - Katolícka univerzita v Ružomberku

Conjecturing: An Investigation Involving Positive Integers 61Although often called Ulam’s Conjecture after the Polish-American mathematicianStanislaw Ulam (1909 - 1984), it was originally stated by theGerman mathematician Lothar Collatz (1910 - 1990). Paul Erdös remarkedthat "Mathematics is not yet ready for such problems." As with Goldbach’sConjecture, it is unproven.It is not true that all conjectures lead to unsolved problems. Now we demonstrateinvestigative approach and creation of conjectures with the help ofthe folloving problem.Problem: Sum of Consecutive Positive IntegersWhat positive integers can be written as a sum of consecutive positiveintegers? For instance, 7 = 3 + 4 and 15 = 4 + 5 + 6, but we cannot write8 as such a sum (try it).Solution:Because we do not know anything about it we must start by using theprocess of experimentation. There are at least two ways of doing this.• The first way is some times known as the heuristic ‚Working backwards’.Take sets of two, then three, then four consecutive numbers, and so on, andfind their sums.For example:1+2 = 3 1+2+3 = 6 1+2+3+4 = 102+3 = 5 2+3+4 = 9 2+3+4+5 = 143+4 = 7 3+4+5 = 12 3+4+5+6 = 18... ... ...This shows us that the integers 3, 5, 7, 6, 9, 12, 10, 14, 18, can be writtenas the sum of consecutive numbers. Let’s order these numbers to see if wecan see a pattern.3, 5, 6, 7, 9, 10, 12, 14, 18, ...If we continue this approach perhaps we might fill in the missing numbersin this sequence or notice a pattern in possible missing numbers. We leaveyou to continue this approach.These are the results of experimenting with the integers 1, 2, 3, 4 and 5:1 = 0+1 Not valid! We must use positive integers and 0 is not positive.2 = We cannot write 2 as the sum of consecutive positive integers.3 = 1+2 Yes!4 = Again, we cannot find a sum.5 = 2+3 Yes!