The integer 7 cannot be expressed as a sum of three or fewer perfect squares. Why does Lagrange's theorem NOT produce a similar exception for sums of four squares?
AThe quaternion norm identity ensures that if m and n are each sums of four squares, so is mn — so the property reduces to primes, and every prime can be shown to admit a four-square representation
BSeven was an oversight in the earlier three-square theorem; four-square sums are simply computed in a wider arithmetic
CAdding a fourth square always converts any non-representable remainder into a perfect square by pigeonhole
DThe four-square theorem only applies to integers greater than 7, sidestepping that case entirely
The key is quaternion norm multiplicativity: N(qr) = N(q)N(r), where N(a+bi+cj+dk) = a²+b²+c²+d². This means the product of two four-square sums is itself a four-square sum. The proof therefore reduces to showing every prime is a sum of four squares — the rest follows from prime factorization. Seven requires four squares but is not a prime-factorization obstacle in the way it would be for a purely multiplicative structure without that identity.
Question 2 Multiple Choice
According to Legendre's three-square theorem, which of the following integers requires all four squares and cannot be written as a sum of three?
A11, since 11 = 8(1) + 3 and fits the pattern
B28, since 28 = 4¹ · 7 = 4¹(8·0 + 7), fitting the form 4ᵃ(8b + 7)
C9, since 9 = 3² uses only one square
D5, since 5 = 4 + 1 uses only two squares
Legendre's three-square theorem states that a positive integer requires four squares (and cannot be written as a sum of three) if and only if it has the form 4ᵃ(8b + 7). For 28: 28 = 4 × 7, and 7 = 8(0) + 7, so 28 = 4¹(8·0 + 7) — exactly this form. For 11: 11 = 9 + 1 + 1 = 3² + 1² + 1², so three squares suffice. For 9 and 5: one and two squares respectively.
Question 3 True / False
The proof that Lagrange's four-square theorem holds for all positive integers reduces to proving it just for prime numbers, because the set of four-square sums is closed under multiplication.
TTrue
FFalse
Answer: True
Quaternion norm multiplicativity is the key algebraic fact: the product of two integers each expressible as sums of four squares is itself expressible as a sum of four squares. This means the property is multiplicative in the same way that prime factorization is. To show every positive integer is a four-square sum, it suffices to show every prime is — the general result follows by combining prime four-square representations using the quaternion identity.
Question 4 True / False
The integers that require most four squares and cannot be written as a sum of three form a finite set known mostly by mathematicians.
TTrue
FFalse
Answer: False
The integers requiring four squares are exactly those of the form 4ᵃ(8b + 7) — an infinite set. For example, 7, 15, 23, 28, 55, 60, 63, 112, … all belong to this family. Legendre's three-square theorem gives a complete characterization, not a finite list. This is what makes Lagrange's result simultaneously tight (four cannot be reduced to three in general) and complete (four always suffices).
Question 5 Short Answer
Why is the multiplicativity of quaternion norms the key step in proving Lagrange's four-square theorem, rather than trying to verify it directly for each integer?
Think about your answer, then reveal below.
Model answer: Quaternion norms satisfy N(qr) = N(q)N(r), where N(a+bi+cj+dk) = a²+b²+c²+d². This means that if two integers are each sums of four squares, their product is also a sum of four squares. Combined with unique prime factorization, this reduces the entire theorem to just one case: proving that every prime is a sum of four squares. A direct case-by-case verification would be impossible (infinitely many integers), but the multiplicativity turns an infinite problem into a single hard case about primes.
This is a classic reduction strategy in number theory. Instead of verifying a property for every integer, you establish a multiplicative closure property and then check the 'atoms' (primes). The quaternion identity provides an explicit algebraic formula: (a²+b²+c²+d²)(e²+f²+g²+h²) = (ae−bf−cg−dh)² + (af+be+ch−dg)² + (ag−bh+ce+df)² + (ah+bg−cf+de)². Euler discovered this identity before quaternions were formalized; Hamilton's quaternion algebra later explained why it works.