Katedra matematiky - Katolícka univerzita v Ružomberku

62 Jan Kopka, Leonard Frobisher, George FeissnerAfter experimenting with the first five integers it is possible to make aconjecture:The experimentation with integers 6 to 11 shows that conjecture 1 is nottrue. For example, 6 and 10 are even integers, and 6 = 1 + 2 + 3 and10 = 1+2+3+4 disprove it. But what about the integer 8? Is it possibleto produce another conjecture knowing that the integers 1,2,4 and 8 cannotbe written as the sum of consecutive integers? What is special about 1,2,4and 8? We know them as powers of 2. Although we have only four of thesespecial numbers we claim:Conjecture 2: Powers of two are cannot be written as the sum of consecutivepositive integers.If we test conjecture 2 with the help of numer 16 = 24, we more confidentthat this conjecture is true. But at this moment we do not know how toconstruct a proof.Now as we can continue our experimentation working systematically withmore and more integers it is possible to say:Conjecture 3: All numbers with the exception of powers of two are sumsof consecutive positive integers.How can we prove conjectures 2 and 3? We must find what distinguishespowers of two from other numbers and then show how this property relatesto the property of being a sum of consecutive numbers.From divisibility theory, we know that the only prime factor of a powerof two is two itself. Thus, all factors of a power of two other than 1 areeven (indeed, they are also powers of two). On the other hand, any positiveinteger not a power of two must have at least one odd factor other than 1.Example:Factors of 2 5 = 32 are: 32,16,8,4,2,1. All factors except 1 are even.Factors of 24 are: 24,12,6,4,3,2,1. There is an odd factor other than 1. Itis number 3.Now we have a distinguishing property, so we can reformulate our conjectures.Conjecture 2a: If a positive integer has only even factors other than 1,then it cannot be written as the sum of consecutive positive integers.Conjecture 3a: If a positive integer has at least one odd factor other than1, then it can be so written.