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The prompt
The prompt
Mathematical inquiry processes: Explore, identify patterns, and generalise; reason and justify. Conceptual field of inquiry: Prime numbers and prime factors; square numbers; modular arithmetic; substitution into a formula; complex numbers.
The prompt is aimed at students in lower secondary school but can be adapted to make it more intriguing for older students. It is in the form of a statement that holds true for only some prime numbers - that is, the number 2 and those in the form 4k + 1 or equivalent to 1 (mod 4).
Students will sometimes interpret the statement as a generalisation that applies to any prime number and declare it to be false based on one counter-example.
In the question, notice, and wonder phase of the inquiry, students have responded to the prompt by defining prime and square numbers before making the following contributions:
To which prime number does the prompt refer?
2 = 12 + 12 . Can you use the same square number?
Three and seven cannot be expressed as the sum of two square numbers.
5 = 12 + 22 works.
As the prime numbers after two are odd, one of the squares has to be odd and the other one is even because Odd + Even = Odd.
The prompt is sometimes true.
Starting the inquiry
Starting the inquiry
When discussing how to proceed, students normally select Find more examples or Make a conjecture from the regulatory cards.
They explain that they want to find more prime numbers that can be expressed as the sum of two square numbers and more that cannot. If they are able to identify the properties of each type, then they can make and explain a conjecture.
The lines of inquiry that develop from the prompt involve contributions to number theory from some of the greatest mathematicians in history, including:
the Brahmagupta-Fibonacci identity;
Fermat's theorem on sums of two squares;
Jacobi's two-square theorem; and
Gauss's norm of complex numbers.
Genesis of the prompt
Genesis of the prompt
The prompt originated in an Inquiry Maths workshop at Dulwich High School in Suzhou, China. A head of mathematics in an international school shared an inquiry prompt he uses with his year 12 (grade 11) classes:
In the prompt two numbers are expressed as the sum of squares in two ways. Finding positive integers that can be expressed as the sum of squares in two or more ways could be a line of inquiry in a structured approach for younger students.
December 2025
Lines of inquiry
Lines of inquiry
The lines of inquiry are shown in more detail in the structured slides.
- Explore prime numbers
When students have split the prime numbers into two categories - those that can be expressed as the sum of two squares and those that cannot - they attempt to make a conjecture. To guide the process, the teacher might point out that many prime numbers come in pairs or connect each list to a sequence (with gaps filled in) and the expression for its nth term.
2. Explore composite numbers
2. Explore composite numbers
The inquiry develops into composite numbers when a student notices that, for example, 10 = 12 + 32.
To determine whether a composite number can be expressed as the sum of two squares requires students to examine the number's prime factors. If the prime factors include a prime number in the form 4k + 3, then it cannot unless the exponent of that prime is even (see the examples of 135 and 405 in the illustration).
3. Express a positive integer as the sum of two squares in two ways
3. Express a positive integer as the sum of two squares in two ways
The teacher might launch this line of inquiry with the first two or three positive integers that can be expressed as the sum of two squares in two ways. Students identify common properties, explain why 55 is not in the list (its prime factors include 11, which is in the form 4k + 3), extend the list, and search for the pairs of squares.
4. How many ways can you express a positive integer as the sum of two squares? How do you find the different ways?
4. How many ways can you express a positive integer as the sum of two squares? How do you find the different ways?
The procedures for working out how many ways a number can be expressed as the sum of two squares and for finding those ways are challenging for younger students.
The teacher might explain an example before students discuss the same example in pairs with 'an explainer' talking aloud and 'a questioner' asking for clarification. The example of 1885 is in the slides and the more complex case of 650 is here.
Students might choose a three-digit number - such as 377, 410, or 425 - and derive the pairs of squares that sum to the number.
5. Find positive integers that can be expressed as the sum of squares using complex numbers
5. Find positive integers that can be expressed as the sum of squares using complex numbers
Students use the following two mathematical facts that connect complex numbers to the sum of two squares to find more cases:
The product of a complex number and its conjugate is the sum of two squares - (a + bi)(a - bi) = a2 + b2.
If two numbers can each be expressed as the sum of two squares, then the product of the number can also be expressed as the sum of two squares.
The example above shows how complex numbers can be used to find pairs of squares that sum to 493. As multiplication is commutative, the complex numbers can be rearranged to create different complex numbers that lead to two squares.
6. Express a positive integer as the sum of two or more cubes
6. Express a positive integer as the sum of two or more cubes
In the final line of inquiry students carry out independent research into positive integers that can be expressed as the sum of two cubes in one or more ways. They might also take the sum of squares inquiry further by researching Legendre's three- and four-square theorems.
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