It is important to explore some of the ideas contained in the film. Today, during class, try some of these exercises introducing Number Theory and explore some exercises designed to introduce Number Theory to you. Show your discussion ideas and your solutions to these exercises in your journal.
Number Theory is partly experimental and partly theoretical. The experimental part normally comes first; it leads to questions and suggests ways to answer them. The theoretical part follows; in this part one tries to devise an argument that gives a conclusive answer to the questions. In summary, here are the steps to follow:
- Accumulate data, usually numerical, but sometimes more abstract in nature.
- Examine the data and try to find patterns and relationships.
- Formulate conjectures (i.e., guesses) that explain the patterns and relation- ships. These are frequently given by formulas.
- Test your conjectures by collecting additional data and checking whether the new information fits your conjectures.
- Devise an argument (i.e., a proof) that your conjectures are correct.
All five steps are important in number theory and in mathematics. More generally, the scientific method always involves at least the first four steps. Be wary of any purported “scientist” who claims to have “proved” something using only the first three. Given any collection of data, it’s generally not too difficult to devise numerous explanations. The true test of a scientific theory is its ability to predict the outcome of experiments that have not yet taken place. In other words, a scientific theory only becomes plausible when it has been tested against new data. This is true of all real science. In mathematics one requires the further step of a proof, that is, a logical sequence of assertions, starting from known facts and ending at the desired statement.
1.1. The first two numbers that are both squares and triangles are 1 and 36. Find the next one and, if possible, the one after that. Can you figure out an efficient way to find triangular–square numbers? Do you think that there are infinitely many?
1.2. Try adding up the first few odd numbers and see if the numbers you get satisfy some sort of pattern. Once you find the pattern, express it as a formula. Give a geometric verification that your formula is correct.
1.3. The consecutive odd numbers 3, 5, and 7 are all primes. Are there infinitely many such “prime triplets”? That is, are there infinitely many prime numbers p such that p + 2 and p + 4 are also primes?
1.4 It is generally believed that infinitely many primes have the form , although no one knows for sure.
- a) Do you think that there are infinitely many primes of the form Nsquared – 1?
- b) Do you think that there are infinitely many primes of the form Nsquared -2?
- c) How about of the form Nsquared -3? How about Nsquared – 4?
- d) Which values of a do you think give infinitely many primes of the form Nsquared – a?
1.5. The following two lines indicate another way to derive the formula for the sum of the first n integers by rearranging the terms in the sum. Fill in the details.
1 + 2 + 3 + · · · + n = (1 + n) + (2 + (n − 1)) + (3 + (n − 2)) + · · · = (1+n)+(1+n)+(1+n)+··· .
How many copies of n + 1 are in there in the second line? You may need to consider the cases of odd n and even n separately. If that’s not clear, first try writing it out explicitly for n = 6 and n = 7.
1.6. For each of the following statements, fill in the blank with an easy-to-check crite- rion:
(a) M is a triangular number if and only if is an odd square.
(b) N is an odd square if and only if is a triangular number.
(c) Prove that your criteria in (a) and (b) are correct.