Can $2018$ be written as a sum of two squares?
If it can be, what are the numbers?
I know the first answer is it can be because $2018=2\times1009$ and $1009\equiv 1 \pmod{4}$. But I cannot find the numbers.
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1If all you care about is the numbers, then wolframalpha can help. – JMoravitz Dec 07 '17 at 04:18
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By hit and trial, you can verify that $2018 = 13^2 + \cdots$ – Math Lover Dec 07 '17 at 04:21
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For a small number like this, just make a column in a spreadsheet with $1$ to $45$ then =sqrt(2018-left^2) will make it easy to find. Just search by eye for the entries without decimals. They leap out. – Ross Millikan Dec 07 '17 at 04:21
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See https://math.stackexchange.com/questions/1828694/algorithm-for-finding-the-representation-of-an-integer-as-a-sum-of-two-squares – lab bhattacharjee Dec 07 '17 at 04:41
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Yes, $2018 = 2\times 1009$ (as prime factorization). Since $1009$ is an odd prime of the form $4k+1$, it can be written as a sum of two squares $1009 = a^2+b^2$. Together with $2 = 1^2 + 1^2$, Brahmagupta–Fibonacci identity $$(a^2+b^2)(c^2+d^2) = (ac-bd)^2 + (ad+bc)^2 = (ac+bd)^2 + (ad-bc)^2$$ tell us $2018 = (a+b)^2 + (a-b)^2$.
The task reduces to rewrite $1009$ as a sum of two squares. In addition to seaching by brute force, one can use algorithms described in answers of this question to determine $a,b$ efficiently. At the end, one find $$1009 = 28^2+15^2 \quad\implies\quad 2018 = 43^2 + 13^2$$
A good reference of these sort of algorithms will be Henri Cohen's book A course in Computation Algebraic Number Theory. Take a look at that if you need more details.
LoPablo
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achille hui
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You are correct my friend,as per Brahmagupta's Fibonacci identity,it says that-$a^2c^2+a^2d^2+b^2c^2+b^2d^2$=$a^2c^2-2acbd+b^2d^2+a^2d^2+2adbc+b^2c^2$=$a^2c^2+2acbd+b^2d^2+a^2d^2-2adbc+b^2c^2$ just as you are writing 1009=784+225 and 2018=1849+169
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Now if you want to use only 43 and 13 to represent 1009 then you can write as follows:$$1009=[(\frac{1}{4}){(43+13)^2+(43-13)^2}]$$= $$(\frac{1}{4}){(56^2+30^2)$$ = $$(\frac{1}{4})(3136+900)$$ – Jan 02 '19 at 05:09