2. Whiteboardmaths.com © 2004 All rights reserved 5 7 2 1

3. 12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 4950515253545556 5758596061626364 6566676869707172 7374757677787980 8182838485868788 8990919293949596 979899100101102103104 Fermat’s Two Square Theorem We can use the sieve of Eratosthenes to catch all primes below 104. The first prime <  104 = 7 Cross out 1 and draw four lines to eliminate multiples of 2, six lines to eliminate multiples of 3, cross out all remaining multiples of 5 and draw two lines to eliminate multiples of 7. Pierre de Fermat (1601 – 1675) 12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 4950515253545556 5758596061626364 6566676869707172 7374757677787980 8182838485868788 8990919293949596 979899100101102103104 Discuss the two colour-coded sets of primes and their divisibility by 4.

4. Prime4 n + 1Prime4n + 3 53 137 1711 2919 3723 4131 5343 6147 7359 8967 9771 10179 83 103 Show that all primes in the left hand column can be written as 4n + 1 and that those in the right hand column can be written as 4n + 3. Prime4 n + 1Prime4n + 3 54 x 1 + 134 x 0 + 3 134 x 3 + 174 x 1 + 3 174 x 4 + 1114 x 2 + 3 294 x 7 + 1194 x 4 + 3 374 x 9 + 1234 x 5 + 3 414 x 10 + 1314 x 7 + 3 534 x 13 + 1434 x 10 + 3 614 x 15 + 1474 x 11 + 3 734 x 18 + 1594 x 14 + 3 894 x 22 + 1674 x 16 + 3 974 x 24 + 1714 x 17 + 3 1014 x 25 + 1794 x 19 + 3 834 x 20 + 3 1034 x 25 + 3 Number12345678910 Square149162536496481100 Using the table of squares below, try to establish a link between the primes of the form 4n + 1and the square numbers. It may take some time. Does this relationship hold for primes of the form 4n + 3? Complete the table to show that all primes below 104 of the form 4n + 1 are the sum of 2 squares.

5. Prime4 n + 1a 2 + b 2 54 x 1 + 1 134 x 3 + 1 174 x 4 + 1 294 x 7 + 1 374 x 9 + 1 414 x 10 + 1 534 x 13 + 1 614 x 15 + 1 734 x 18 + 1 894 x 22 + 1 974 x 24 + 1 1014 x 25 + 1 Number12345678910 Square149162536496481100 Show that all primes below 104 of the form 4n + 1 are the sum of 2 squares. Prime4 n + 1a 2 + b 2 54 x 1 + 11 2 + 2 2 134 x 3 + 12 2 + 3 2 174 x 4 + 11 2 + 4 2 294 x 7 + 12 2 + 5 2 374 x 9 + 11 2 + 6 2 414 x 10 + 14 2 + 5 2 534 x 13 + 12 2 + 7 2 614 x 15 + 15 2 + 6 2 734 x 18 + 13 2 + 8 2 894 x 22 + 15 2 + 8 2 974 x 24 + 14 2 + 9 2 1014 x 25 + 11 2 + 10 2 Fermat’s 2 Square Theorem All primes of the form 4n + 1 are the sum of 2 Squares The Theorem is extremely difficult to prove by anyone other than an expert mathematician. However it is much easier to show that primes of the form 4n + 3 can never be written as the sum of 2 squares. Can you state Fermat’s Theorem?

6. Prime4n + 3 34 x 0 + 3 74 x 1 + 3 114 x 2 + 3 194 x 4 + 3 234 x 5 + 3 314 x 7 + 3 434 x 10 + 3 474 x 11 + 3 594 x 14 + 3 674 x 16 + 3 714 x 17 + 3 794 x 19 + 3 834 x 20 + 3 1034 x 25 + 3 Number12345678910 Square149162536496481100 Try to show that a 2 + b 2 can never be of the form 4n + 3 by considering even and odd values of a and b together with the remainder after division by 4. Remember: an even number is of the form 2m for some m, and an odd number is of the form 2k + 1, for some k and a multiple of 4 of the form 4q for some q. Case1: both a and b even (2m) 2 + (2k) 2 = 4m 2 + 4k 2 = 4(m + k) Case2: one of them even and one of them odd (2m) 2 + (2k + 1) 2 = 4m 2 +4k 2 +4k + 1= 4(m 2 +k 2 + k) + 1 Case3: both odd (2m + 1) 2 + (2k + 1) 2 = 4m 2 + 4m + 1 + 4k 2 +4k + 1 = 4(m 2 + k 2 +m + k) + 2 In all possible cases the remainder on division by 4 is either 0, 1 or 2. Thus numbers of the form 4n + 3 can never be expressed as the sum of two squares.

7. Worksheet 1 12345678 910111213141516 1718192021222324 2526272829303132 3334353637383940 4142434445464748 4950515253545556 5758596061626364 6566676869707172 7374757677787980 8182838485868788 8990919293949596 979899100101102103104

8. Worksheet 2 Prime4 n + 1Prime4n + 3 53 137 1711 2919 3723 4131 5343 6147 7359 8967 9771 10179 83 103 Number12345678910 Square149162536496481100

9. Worksheet 3 Prime4 n + 1a 2 + b 2 54 x 1 + 1 134 x 3 + 1 174 x 4 + 1 294 x 7 + 1 374 x 9 + 1 414 x 10 + 1 534 x 13 + 1 614 x 15 + 1 734 x 18 + 1 894 x 22 + 1 974 x 24 + 1 1014 x 25 + 1 Number12345678910 Square149162536496481100