Sums of Squares II. Which numbers are sums of two squares? It often turns out that questions of this sort are easier to answer first
Saturday, December 26, 2015
Around and around you are spinning me..............
Sums of Squares II. Which numbers are sums of two squares? It often turns out that questions of this sort are easier to answer firstfor primes, so we ask
which (odd) prime numbers are a sum of two squares. For example,
3 = NO, 5 = 12 + 22
, 7 = NO, 11 = NO,
13 = 22 + 32
, 17 = 12 + 42
, 19 = NO, 23 = NO,
29 = 22 + 52
, 31 = NO, 37 = 12 + 62
, . . .
Do you see a pattern? Possibly not, since this is only a short list, but a longer
list leads to the conjecture that p is a sum of two squares if it is congruent
to 1 (modulo 4). In other words, p is a sum of two squares if it leaves a
remainder of 1 when divided by 4, and it is not a sum of two squares if it
leaves a remainder of 3. We will prove that this is true in Chapter 24.
Sums of Squares II. Which numbers are sums of two squares? It often turns out that questions of this sort are easier to answer first
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