Download Fermat`s 2 Square Theorem

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Fermat’s Two Square Theorem
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We can use the sieve of
Eratosthenes to catch all
primes below 104.
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The first prime < 104 = 7
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Pierre de Fermat
(1601 – 1675)
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.
Discuss the two colour-coded sets of primes and their divisibility by 4.
Prime
4n+1
Prime
4n + 3
5
4x1+1
3
4x0+3
13
4x3+1
7
4x1+3
17
4x4+1
11
4x2+3
29
4x7+1
19
4x4+3
37
4x9+1
23
4x5+3
41
4 x 10 + 1
31
4x7+3
53
4 x 13 + 1
43
4 x 10 + 3
61
4 x 15 + 1
47
4 x 11 + 3
73
4 x 18 + 1
59
4 x 14 + 3
89
4 x 22 + 1
67
4 x 16 + 3
97
4 x 24 + 1
71
4 x 17 + 3
101
4 x 25 + 1
79
4 x 19 + 3
83
4 x 20 + 3
103
4 x 25 + 3
Number
Square
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.
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.
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Prime
4n+1
a2 + b2
5
4x1+1
12 + 2 2
13
4x3+1
22 + 3 2
17
4x4+1
12 + 4 2
29
4x7+1
22 + 5 2
37
4x9+1
12
41
4 x 10 + 1
42 + 52
53
4 x 13 + 1
22 + 7 2
61
4 x 15 + 1
52 + 6 2
73
4 x 18 + 1
32 + 8 2
89
4 x 22 + 1
52 + 8 2
97
4 x 24 + 1
42 + 92
101
4 x 25 + 1
12 + 102
Number
Square
+
Show that all primes below 104 of the form
4n + 1 are the sum of 2 squares.
Can you state Fermat’s Theorem?
Fermat’s 2 Square Theorem
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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.
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Try to show that a2 + b2 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.
Prime
4n + 3
3
4x0+3
7
4x1+3
11
4x2+3
19
4x4+3
23
4x5+3
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4x7+3
Case1: both a and b even
(2m)2 + (2k)2 = 4m2 + 4k2 = 4(m + k)
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4 x 10 + 3
Case2: one of them even and one of them odd
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4 x 11 + 3
(2m)2 + (2k + 1)2 = 4m2 +4k2 +4k + 1= 4(m2 +k2 + k) + 1
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4 x 14 + 3
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4 x 16 + 3
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4 x 17 + 3
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4 x 19 + 3
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4 x 20 + 3
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4 x 25 + 3
Number
Square
Case3: both odd
(2m + 1)2 + (2k + 1)2 = 4m2 + 4m + 1 + 4k2 +4k + 1
= 4(m2 + k2 +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.
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Worksheet 1
Worksheet 2
Prime
4n+1
Prime
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101
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4n + 3
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103
Number
Square
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Worksheet 3
Number
Square
Prime
4n+1
5
4x1+1
13
4x3+1
17
4x4+1
29
4x7+1
37
4x9+1
41
4 x 10 + 1
53
4 x 13 + 1
61
4 x 15 + 1
73
4 x 18 + 1
89
4 x 22 + 1
97
4 x 24 + 1
101
4 x 25 + 1
a2 + b2
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