Number theory through inquiry
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The Heart of Mathematics: An invitation to effective thinking The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking.
Starbird is working with two Ph. Thus, the notes for those topics will be updated soon. Please remember that these versions are just drafts and therefore have errors and weaknesses that will be fixed. If r1, r2,.
To prove the following theorem, remember the proof of the Infinitude of Primes Theorem and see how the strategy of that proof might be adapted to prove the following harder theorem. There are infinitely many prime numbers that are congruent to 3 modulo 4. When you have proved the previous theorem, you will have forced yourself to understand a technique of proving theorems about the existence of infinitely many primes of a certain type. Now is the time to see how far that technique can be pushed. In other words ask yourself how many theorems like the preceding one are provable using a similar idea.
Are there other theorems like the previous one that you can prove? In fact, the following much more general theorem is true. Its proof in its full generality, however, is quite difficult and we will not attempt it in this course. It is a sequence of numbers all of which are congruent to b modulo a. The study of primes in arithmetic progressions is still an active field today. Consider the following recent result due to Ben Green and Terrence Tao. Theorem Green and Tao, There are arbitrarily long arithmetic progressions of primes. Terrence Tao was awarded a Fields medal in part for his work related to this result.
Fields medals, the mathematical equivalent of the Nobel prize, are awarded once every four years to outstanding mathematicians under the age of Find the current record for the longest arithmetic progression of primes. Primes of special form.
The theorems here show some features of Mersenne primes and related primes. Find the first few Mersenne primes and Fermat primes. For an A in the class and a Ph. The distribution of primes.
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How are the primes distributed among the natural numbers? Is there some pattern to their distribution? There are infinitely many primes, but how rare are they among the numbers? What proportion of the natural numbers are prime numbers? To explore these questions, the best way to start is to look at the natural numbers and the primes among them. Here then are some ranges of natural numbers with the primes printed in bold: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24,.
What observations can we make? First, we may notice that the proportion of bold numbers occuring seems to be getting smaller. That is, primes tend to be more sparse as we move further out into the sequence of natural numbers. Stated another way, we tend to see longer and longer runs of consecutive composite numbers. There exist arbitrarily long strings of consecutive composite numbers. That is, for any natural number n there is a string of more than n consecutive composite numbers.
Number theory through inquiry | UTS Library
On the other hand, we still observe primes clustered together, such as and , or and It is a famous open question as to whether or not this behavior continues indefinitely. If you have already settled the previous question about Mersenne primes, then solving the following question will give you another Ph.
Question The Twin Primes Question. Are there infinitely many pairs of prime numbers that differ from one another by two? The pairs 11 and 13, 29 and 31, 41 and 43 are examples of some such pairs. Out of the first 24 natural numbers, 9 of them are primes. We saw how this fraction changes as n increases in the Sieve of Eratosthenes exercise.
Table 1. Prime Proportions Before high-speed computers were available, calculating or just estimating the proportion of prime numbers in the natural numbers was a difficult task. An eighteenth-century Austrian arithmetician by the name of J. Kulik spent 20 years of his life creating, by hand, a table of the first million primes. His table was never published and sadly the volume containing the primes between 12,, and 22,, has since disappeared. As we observed above, the proportion of primes seems to slowly go downward.
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That is, the percentage of numbers less than a million that are prime is smaller than the percentage of numbers less than a thousand that are prime. The primes, in some sense, get sparser and sparser among the bigger numbers.
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They noticed that even though primes do not appear to occur in any predictable pattern, the proportion of primes is related to the natural logarithm. Gauss and Legendre conjectured that the proportion of primes among the first n natural numbers is approximately 1 ln n. Table 1 shows the number of primes up to n, the proportions of primes, and a comparison with 1 ln n. Theorem The Prime Number Theorem. Finally, we mention here one more famous open question concerning prime numbers.
Express each of the first 20 even numbers greater than 2 as a sum of two primes. In a letter to Euler, dated June 7, , Christian Goldbach claimed that every natural number greater than 2 was the sum of three primes. It was convention at the time to include the number 1 as being among the primes. The conjecture was re-expressed by Euler as follows. Conjecture The Goldbach Conjecture. Every positive, even number greater than 2 can be written as the sum of two primes.
The Goldbach Conjecture has been verified by computer, as of June of , for all even numbers up to ,,,,, As the even numbers get larger, there seem to be more ways to write them as a sum of two primes. For example, the number ,, can be written as the sum of two primes in , different ways.
But no one knows how to prove that in general all even natural numbers are the sum of two primes. Perhaps some even number with 10 trillion digits is not the sum of two primes. Until we have a general method of proof that will apply to all even numbers, we will not know whether such a natural number might not exist.
For example, Proposition 20 of Book IX gives the first known proof of the infinitude of primes. A natural number is said to be perfect if it is equal to the sum of its proper divisors. The serious study of perfect numbers and primes of special forms was picked up again in the seventeenth century by the likes of Rene Descartes , Pierre de Fermat , and Marin Mersenne He erred only by including 67 and excluding 61, 89 and Curiously, it is not known if any odd perfect numbers exist.
The search for new Mersenne primes continues to this day. Find the current record for the largest known Mersenne prime. So happy hunting. Now we proceed to see how modular arithmetic relates to other familiar algebraic constructions such as functions and equations, and how it can help us to better understand primes and composite numbers.
Modular arithmetic is interesting as an abstract topic in number theory, but it also plays important roles in real life.
It is the basis for public key cryptography and check digits associated with error detection.