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The Gaussian Primes

The Gaussian Primes. Tony Shaheen CSU Los Angeles. This talk can be found on my website: www.calstatela.edu/faculty/ashahee/. These are the Gaussian primes. The picture is from http ://mathworld.wolfram.com/GaussianPrime.html.

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The Gaussian Primes

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  1. The Gaussian Primes Tony Shaheen CSU Los Angeles This talk can be found on my website: www.calstatela.edu/faculty/ashahee/

  2. These are the Gaussian primes. The picture is from http://mathworld.wolfram.com/GaussianPrime.html

  3. Do you think you can start near the middle and jump along the dots with jumps of bounded size (like size 3) and make it out to infinity? That is, are there arbitrarily large gaps in the Gaussian primes?

  4. No one knows the answer to this question. We do know the answer for the ordinary integers though. The answer there is no. This talk is about these two problems.

  5. The set of positive and negative whole numbers, including 0, is called the set of integers. We denote this set by the symbol . That is,

  6. We can visualize the integers as an infinite set of points on the number line extending in both directions. -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 0 1

  7. Let be an integer. We say that x is a unit if 1/x is an integer. For example, 1 and -1 are units since is an integer and is an integer. But 2 is not a unit since is not an integer. That is, we cannot divide by 2 in the land of integers.

  8. The only units in are 1 and -1. -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 0 1

  9. Units allow one to create factorizations of an integer that don’t really decompose the integer into smaller pieces. For example: 5 = 1 5 = 151 = (-1)(-1) 5 = (-1) (-1) (1) (-1) (-1) 5

  10. Let be a integer where p is not a unit. We say that p is a prime element if the following is true: For all integers x and y, if p = x y, then either x or y is a unit.

  11. Let be a integer where p is not a unit. We say that p is a prime element if the following is true: For all integers x and y, if p = x y, then either x or y is a unit. Note: We are allowing negative integers to be prime elements. This differs from the standard definition of prime, but will be match with the definition for the Gaussian integers. There is a more general definition for any commutative ring: p is prime if whenever p | xy we have p | x or p | y.

  12. Ex: Consider the integer n = 6. 6 is not prime because 6 = 23 and neither 2 nor 3 is a unit. We could have also written 6 = (-2)3)

  13. Ex: Consider the integer n = 7. The only way to write 7 = xy with integers x and y is 7 = 7 1 7 = 1 7 7 = (-7) (-1) 7 = (-1) (-7) and in all these cases one of the factors is a unit. Hence 7 is a prime element in the integers. Similarly -7 is a prime element.

  14. Let be an integer. Note that So every non-zero integer has the following divisors: 1, -1, x, and –x. Another way to think of a prime integer is an integer whose only divisors are 1, -1, x, and –x.

  15. Here are some of the prime elements in the integers. Note the symmetry about the integer 0. -7 -6 -5 -4 -3 -2 -1 2 3 4 5 6 7 0 1

  16. Question: Can one start at the prime 2 and then jump along the primes to infinity where the jumps are of bounded length? 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, …

  17. For example, suppose we start at 2 and can only make jumps of size at most 7. Then we get stopped at the prime 89. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, …

  18. What about steps of size 100,000? What about steps of size 1,000,000? Are there arbitrarily large gaps in the primes?

  19. What about steps of size 100,000? What about steps of size 1,000,000? Are there arbitrarily large gaps in the primes? For the integers we can answer this question.

  20. Let N be a positive integer. Here N represents the jump size. Consider the following list of consecutive integers: (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1)

  21. Let N be a positive integer. Here N represents the jump size. Consider the following list of consecutive integers: (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1) For example, if N = 4, then we have the sequence: 5! + 2, 5! + 3, 5! + 4, 5! + 5

  22. Let N be a positive integer. Here N represents the jump size. Consider the following list of consecutive integers: (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1) For example, if N = 4, then we have the sequence: 5! + 2, 5! + 3, 5! + 4, 5! + 5 , , (,

  23. Let N be a positive integer. Here N represents the jump size. Consider the following list of consecutive integers: (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1) For example, if N = 4, then we have the sequence: 5! + 2, 5! + 3, 5! + 4, 5! + 5 , , (, ,, (,

  24. Let N be a positive integer. Here N represents the jump size. Consider the following list of consecutive integers: (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1) For example, if N = 4, then we have the sequence: 5! + 2, 5! + 3, 5! + 4, 5! + 5 , , (, ,, (, 2(61), 3(41), 4(31), 5(25) 122, 123, 124, 125

  25. (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1)

  26. (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1) In general, (N+1)! + k is composite since (N+1)! + k is divisible by k.

  27. (N+1)! + 2, (N+1)! + 3, (N+1)! + 4, …, (N+1)! + (N+1) In general, (N+1)! + k is composite since (N+1)! + k is divisible by k. Hence, for any positive integer N we can make a list of N consecutive integers where none of them are prime.

  28. So we cannot start at 2 and jump along the primes to infinity with jumps of size N = 10

  29. So we cannot start at 2 and jump along the primes to infinity with jumps of size N = 10 or N = 10,000

  30. So we cannot start at 2 and jump along the primes to infinity with jumps of size N = 10 or N = 10,000 or N = 10,000,000,000,000,000,000,000,000,000,000

  31. So we cannot start at 2 and jump along the primes to infinity with jumps of size N = 10 or N = 10,000 or N = 10,000,000,000,000,000,000,000,000,000,000 or any N.

  32. Hence the primes contain arbitrarily large gaps.

  33. What about the Gaussian primes?

  34. What about the Gaussian primes? Before we introduce the Gaussian primes we need to know something about the integer primes.

  35. The positive primes break into 3 groups.

  36. The positive primes break into 3 groups. There is a theorem of Dirichlet that implies that the sets and are both infinite in size. http://en.wikipedia.org/wiki/Dirichlet's_theorem_on_arithmetic_progressions

  37. Question: Given a positive prime p, do there exist integers x and y with ? We say that p expressible as a sum of squares if p is of the above form.

  38. 2 is prime and is a sum of squares.

  39. 2 is prime and is a sum of squares. 5 is prime and is a sum of squares. Notice that 5

  40. 2 is prime and is a sum of squares. 5 is prime and is a sum of squares. Notice that 5 13 is prime and is a sum of squares. Notice that 13

  41. 2 is prime and is a sum of squares. 5 is prime and is a sum of squares. Notice that 5 13 is prime and is a sum of squares. Notice that 13 17 is prime and is a sum of squares. Notice that 17

  42. 2 is prime and is a sum of squares. 5 is prime and is a sum of squares. Notice that 5 13 is prime and is a sum of squares. Notice that 13 17 is prime and is a sum of squares. Notice that 17 29 is prime and is a sum of squares. Notice that 29

  43. 2 is prime and is a sum of squares. 5 is prime and is a sum of squares. Notice that 5 13 is prime and is a sum of squares. Notice that 13 17 is prime and is a sum of squares. Notice that 17 29 is prime and is a sum of squares. Notice that 29 41 is prime and is a sum of squares. Notice that 41

  44. Notice that we haven’t found any positive primes p with p that are the sum of squares.

  45. Notice that we haven’t found any positive primes p with p that are the sum of squares. That’s because there aren’t any.

  46. Theorem: A positive prime p is expressible as a sum of squares if and only if = 2

  47. Theorem: A positive prime p is expressible as a sum of squaresif and only if = 2 According to wikipedia: Albert Girard (1632) was the first to make this observation. Fermat claimed to have a proof of it in a letter to Mersenne (1640). Euler was the first person to provide a proof in 1749. There have been many different proofs of the above theorem. http://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares

  48. The following set is called the set of Gaussian integers. -3+ 2i 3+ i i 1+ i 2+ i -5 -4 -3 -2 -1 2 3 4 5 0 1 -2i 4 - 2i -2- 3i

  49. Let z = a + bi be a Gaussian integer. The norm of z is

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