Some numbers can be expressed as the difference of two perfect squares:
$20 = 6^2 - 4^2$
$21 = 5^2 - 2^2$
$36 = 6^2 - 0^2$
$165 = 13^2-2^2$
How many of the numbers from $1$ to $30$ can you express as the difference of two perfect squares?
Here are some questions you might like to consider:
What is special about the difference between squares of consecutive numbers? Why?
What about the difference between the squares of two numbers which differ by $2$? By $3$? By $4$...?
When is the difference between two square numbers odd?
And when is it even?
What do you notice about the numbers you CANNOT express as the difference of two perfect squares?
Can you prove any of your findings?
Comments
5, 7, 9
the sequence of the squares $x^2 -2^2 = 5, 12, 21, 32$ and the difference between those = $5, 7, 9, 11$
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What's Possible?
Question: How many of the numbers from 1 to 30 can you express as the difference of two perfect squares?
Answer: 13 numbers: 3, 5, 7, 8, 9, 11, 12, 15, 16, 20, 21, 24 and 27.
Proof:
a) 2, 3, 4 and 5 squared, each one minus 1² = 3, 8, 15 and 24
b) 3, 4 and 5 squared, each one minus 2² = 5, 12 and 21
c) 4, 5 and 6 squared, each one minus 3² = 7, 16 and 27
d) 5 and 6 squared, each one minus 4² = 5 and 20
e) lastly, 6² minus 5² = 11
Q.: What is special about the difference between squares of consecutive numbers? Why?
A.: If we take case (a) from the above answer (2,3,4 & 5 squared, each one minus 1²...) the result of the first binomial (2² - 1²) is 3.
If we take the second case (b), the result of the first binomial of the list (3² - 2²) will be 5.
Taking the third case (c), result = 7.
Taking the fourth case (d), result = 9.
The next will be 11.
Therefore, if we carry on indefinitely, the following results will ALWAYS come in increments of two and they will ALWAYS be ODD NUMBERS, never EVEN (3, 5, 7, 9, 11, 13, 15, 17, 19, 21,...).
Q.: What about the difference between the squares of two numbers which differ by 2 ? By 3? By 4 ...?
A.: Let’s examine examples:
(i) 3² - 1² = 8 (both terms differ by 2)
(ii) 4² - 2² = 12 (both by 2)
(iii) 5² - 3² = 16 (both by 2)
(iv) 6² - 4² = 20 (both by 2)
(v) 4² - 1² = 15 (both terms differ by 3)
(vi) 5² - 2² = 21 (by 3)
(vii) 6² - 3² = 27 (by 3)
(viii) 7² - 4² = 33 (by 3)
(ix) 5² - 1² = 24 (both terms differ by 4)
(x) 6² - 2² = 32 (by 4)
(xi) 7² - 3² = 40 (by 4)
(xii) 8² - 4² = 48 (by 4)
(xiii) 6² - 1² = 35 (both terms differ by 5)
(xiv) 7² - 2² = 45 (by 5)
(xv) 8² - 3² = 55 (by 5)
(xvi) 9² - 4² = 65 (by 5)
We observe the following sequences of numbers:
a) When they differ by 2 (i to iv): 8, 12, 16, 20,... i.e. increments of 4
b) When they differ by 3 (v to viii): 15, 21, 27, 33,... i.e. increments of 6
c) When they differ by 4 (ix to xii): 24, 32, 40, 48,... i.e. increments of 8
d) When they differ by 5 (xiv to xvi): 35, 45, 55, 65,... i.e. increments of 10
We may therefore infer a simple rule: “The difference between the squares of two numbers which differ consecutively by 2, 3, 4, 5,... n will always be a sequence of even numbers 4, 6, 8, 10, 12, 14,... n. That is to say, the resulting sequence will always be the double of the difference between the squares of two numbers”.
Q.: When is the difference between two square numbers odd? And when is it even?
A.: a) If 1st term is EVEN and 2nd term is ODD, then the result is ALWAYS ODD
b) If 1st term is ODD and 2nd term is EVEN, then the result is ALWAYS ODD
c) If both terms are EVEN, then the result will ALWAYS be EVEN
d) If both terms are ODD, then the the result will ALWAYS be EVEN
Q.: What do you notice about the numbers you CANNOT express as the difference of two perfect squares?
A.: If you CANNOT express a number RESULT as the difference of two perfect squares, there is NO RESULT. Therefore we have an impossible binomial which renders an impossible result. Therefore, as there is no result, we may notice NOTHING at all about numbers we cannot express as the difference of two perfect squares.
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