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Question 448561: Which of the following CANT be expressed as the sum of the squares of two integers? Is there any formula to solve such problems?
(a)13
(b)17
(c)21
(d)29
(e)34
Found 2 solutions by solver91311, Edwin McCravy:
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The first 5 perfect squares are:
1, 4, 9, 16, and 25.
You won't need the 6th one because it is larger than the largest of your choices.
Start with your first one, 13.
13 minus 1 is 12. 1 is a perfect square, 12 is not.
13 minus 4 is 9. 4 is a perfect square, 9 is a perfect square, so 13 is the sum of two perfect squares. Eliminate this choice.
17 minus 1 is 16. 1 and 16 are perfect squares. Eliminate choice.
Keep going until you find one that is not the sum of two perfect squares. I can say that there is only one in the given list of numbers, but I wouldn't trust me if I were you -- check them all.
By the way, the word "can't" is properly spelled with an apostrophe.
John
My calculator said it, I believe it, that settles it
Answer by Edwin McCravy(20060)  (Show Source):
You can put this solution on YOUR website!
The other tutor didn't give a formula but there is one.
Euler proved that a positive integer N is expressible as the sum
of the squares of two different integers if and only if in the
prime factorization of N, every prime number of the form (4k+3),
where k is a positive integer, occurs an even number of times.
Note that 0 is an even number, so if a prime of the form (4k+3),
where k is a positive integer, doesn't occur at all in the prime
factorization, then it is considered as occuring an even number
of times.
The numbers of the form 4k+3 are
k 4k+3
1 7
2 11
3 15, not prime
4 19
5 23
6 27, not prime
7 31
8 35, not prime
So here is the list of primes of the form 4k+3
which are less than the numbers in your problem:
7,11,19,23,31
--------------------------
(a) 13
The prime factorization of 13 is just 13 which contains
all those in the list 0 times, and 0 is an even number,
so 13 can be written as the sum of two squares of integers,
2²+3²=4+9=13
(b) 17
The prime factorization of 17 is just 17 which contains
all those in the list 0 times, and 0 is an even number,
so 17 can be written as the sum of two squares of integers,
1²+4²=1+16=17
(c) 21
The prime factorization of 21 is 3*7 which contains
7 one time, and 1 is an odd number, so 21 cannot
be written as the sum of two squares of integers.
(d) 29
The prime factorization of 29 is just 29 which contains
all those in the list 0 times, and 0 is an even number,
so 29 can be written as the sum of two squares of integers,
2²+5²=4+25=29
(e) 34
The prime factorization of 34 is just 2*17, but neither 2
nor 17 are in the list of primes of the form 4k+3, so they
don't matter. So 34's prime factorization contains all
those in the list 0 times, and 0 is an even number, so 34
can be written as the sum of two squares of integers,
3²+5²=9+25=34
Edwin
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