Chapter 4 Elementary Number Theory

1. Chapter 4 Elementary Number Theory Unless otherwise specified, the domain of any predicate will be the set of natural numbers N = {0, 1, 2, 3, …} The non-logical symbols that we can use will be +, ×, >, =, 0, 1

2. Basic Definitions • the number a divides b iff •  c (b = a × c) • the notation is a | b • a number n is even iff it is divisible by 2 •  k (n = 2 × k) • a number n is odd iff •  k (n = 2 × k + 1) • a number p is prime iff • (p > 1)  [a,b (a × b = p  a = 1  b = 1)]

3. Pierre Fermat (1601 – 1665) Next to the discussion of the equation x2 + y2 = z2 in his copy of the works of Diophantus, Fermat wrote in the margin: “However, it is impossible to write a cube as the sum of two cubes, a 4th power as the sum of two 4th powers and in general any power the sum of two similar powers. For this I have discovered a truly wonderful proof, but this margin is too small to contain it.” Fermat gave a proof for the case n = 4 in 1659, Euler gave a proof for the case n = 3 in 1770. Legendre gave a proof for the case n = 5 in 1823. Andrew Wiles finally gave a complete proof for this theorem in 1995 (the first proof he gave in 1993 has a gap in it). Translating English into 1st order logic • Examples: • n is the sum of two squares • a b (n = a×a + b×b) • there are infinitely many pairs of twin primes •  n p ( p>n  “p is prime”  “p+2 is prime”) • Fermat’s last theorem: for any integer n > 2, there is no • positive integer solution to the equation x n + y n = z n • n>2 [x>0 y>0 z>0 (x n + y n = z n)] • c is the GCD of a and b • {s t (a = c×s  b = c×t)}  • d [s t (a = d×s  b = d×t)  d  c]

4. Why do we need a proof? Conjecture: If p is prime, then 2p – 1 is also prime. It is easy to see that 22 – 1 = 3 is prime, 23 – 1 = 7, and 25 – 1= 31 are all primes, so one may think that this conjecture is really true. Actually, 211 – 1 = 2047 = 23 × 89 is not prime. Note: Any prime of the form 2p – 1 is called a Mersenne prime, the largest one up to date is 225,964,951 – 1 (discovered on 2-18-05)

5. Why do we need a proof? Fermat conjecture that: For every whole number n, Fn = 2(2n) + 1 is prime. It is easy to see that F0 = 3 is prime, F1 = 5, and F2 = 17, F3 = 257, F4 = 65537 are all primes. But F5 is too large for Fermat to factorize it, hence he conjectured that this formula always produces prime. Actually, Euler proved (a century later) that F5 = 232 +1 = 641 × 6,700,417 is not prime. Note: Up to 2005, there is no other Fermat prime found above 65537.

6. Twin Primes The largest discovered to date is the pair 4,648,619,711,505×260,000 1 These two primes each have 18,075 digits. They were discovered in the year 2000 using a super computer.

7. Basic types of proofs • Constructive proof of existential statements • a concrete example is constructed • Example: prove that the equation x2 + y2 = z2 has positive integer solutions • Proof: Consider the integers 3, 4, and 5. • We see that 32 + 42 = 25 = 52 ,therefore the • statement is true. Q.E.D. • Q.E.D. means quod erat demonstrandum in Latin, • in English this means “this is what needed to be • shown”

8. Example: Prove that there are distinct integers a and b such that ab = ba Proof: Consider the integers 2, and 4. We see that 24 =16 and 42 = 16,therefore the statement is true. Q.E.D. Example: Prove that there are integers a and b such that 7a + 12b = GCD(7, 12) Proof: Consider … … Q.E.D.

9. Another example of Constructive proof. For more than 200 years the Mersenne number 267 -1 was considered to be prime. In 1903, Frank Nelson Cole, in a speech to the American Mathematical Society, went to the blackboard and without uttering a word, raised 2 to the power 67 by hand (using our usual multiplication algorithm!) and subtract 1. He then multiply 193,707,721 by 761,838,257,287 also by hand. The two numbers matched! His audience greeted the presentation with a standing ovation. When asked how long to took him to crack the number, he said, “All the Sundays in 3 years.”

10. Non-constructive prove of existence • this is usually accomplished by a cardinality argument, • namely to show that AB, we can first establish that • A  B and then show that B actually has more elements • than A. Example: Prove that there is a non-algebraic real number. (a non-algebraic number is also called a transcendental number.) Definition: A number (real or complex) is said to be algebraic if it is a solution of a polynomial equation with integer coefficients. eg. 2 is a solution to the equation x2 – 2 = 0 (1+ 5)2 is a solution to the equation x2 – x – 1 =0

11. Clearly every rational number is algebraic. The following picture summarizes the situation. We can show that there are only countably many real algebraic numbers but there are uncountably many real numbers, therefore there must be some (in fact quite a lot) of transcendental numbers.

12. Even though we knew that there are infinitely many transcendental numbers, it took mathematicians many years to actually find an example: Liouville's number (1844) 0.11000100000000000000000100000000 ... which has a one in the 1st, 2nd, 6th, 24th, etc. places and zeros elsewhere. Chapernowne's number, 0.12345678910111213141516171819202122232425... This is constructed by concatenating the digits of the positive integers. (Can you see the pattern?) Note: e and  are shown to be transcendental much later.

13. Disproving Universal Statements by Counterexample To disprove a statement of the form “ xD, if P(x) then Q(x)” find a value of x in D such that Q(x) is false. Such an x is called a counterexample. Example: Statement: for every real numbers a and b, if a2 = b2, then a = b. Disproof: Since (-2)2 = 4 and 22 = 4, but -2  2, therefore the above statement is false.

14. Method of exhaustion for universal statements • works only for a finite domain • need to check every single element in the domain • Example: For every 3-digit number n, if the sum of the first and • last digit equals to the middle digit, then n is divisible • by 11. • Proof: n = 110 is divisible by 11, • n = 121 is divisible by 11, • n = 132 is divisible by 11, • … • n = 220 is divisible by 11, • n = 231 is divisible by 11, • n = 242 is divisible by 11, • … • n = 990 is divisible by 11. Q.E.D.

15. Method of exhaustion for universal statements This method fails for infinite domains. However, quite often we would still like to use brute force to check the validity of the statement for a finite subset of the domain. The experience we get from there will be very valuable to the construction a real proof. Example: Goldbach’s conjecture (1742) Every even integer n> 2 is a sum of two primes. Verify all even numbers n ≤ 30 . 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, …

16. Goldbach’s conjecture (1742 ) Up to Feb 1, 2005, all even numbers up to 2×1017 have been checked and they all can be written as a sum of two primes. However, no body can prove this conjecture yet. The best result so far is (Chen 1973) Every sufficiently large even number is either the sum of two odd primes or the sum of an odd prime and a product of two primes. Note: it is also shown that every sufficiently large odd number is a sum of three primes. The lower bound is 107194

17. The most controversial application of the exhaustion method. Four Color Theorem Any map on the plane can be painted by 4 colors so that no two adjacent regions have the same color. The domain of this problem is clearly infinite, but mathematicians managed to classify the infinite set of maps into around 1500 different types. After that, they used 1200 hours of computer time to check all these different types of maps and concluded that this conjecture is really correct (1989). For a detail description, check www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_four_colour_theorem.html

18. Method of Generalizing from the generic particular. • the most powerful and commonly used method to prove an universal statement. • An object is said to be a generic particular if it is randomly chosen from the domain but shares the characteristics common to all elements in the domain. • Example: Prove that the product of an even number and an • odd number is even. Example: Prove that any odd prime can be written as the difference of two squares in one and only one way.

19. The power function y = x n can be expressed in terms of +, × and logical symbols. where (c, a, k) is the Gödel -function defined by clearly

20. Chinese Remainder Theorem Let m1, m2, m3, … mk be pairwise relatively prime natural numbers. Then the system of congruence x a1 (mod m1) x a2 (mod m2) ….. x ak (mod mk) always has a solution. Corollary Given any finite sequence of natural numbers r1, r2, …, rn, there are natural numbers c and a such that for every i from 1 to n, (c, a, i) = ri

21. proof: Let s = max{n, r1, r2, …, rn} + 1, and a = s! Then ai +1 > ri for all i and {a+1, 2a+1, …, na+1} are pairwise relatively prime, hence c exists by the Chinese remainder theorem.