It is important to explore some of the ideas contained in the film we will be watching. Today, during the first part of class, try some of these exercises introducing Number Theory and explore some exercises designed to introduce Number Theory to you.
Number Theory 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:
Exercises 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.
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. Today we worked for 15 minutes on the above! We turned in the final index and then watched the first 35’ish minutes of the movie Pi (1998) by Darren Aronofsky. If you were absent, please work on the above for 15 minutes and then watch the movie. You can find it on your own or make an appointment to view it with me after school one day. Additionally, please turn in your final index when you return to class. Important Dates 1/22: Final Binder Check at the start of class – even if you are absent; will be marked down 15% per day from the start of class period (rubrics handed out in class Jan 8th; entries will include where we left of from Journal Check 2 to Journal #27) Comments are closed.
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