Minkowski's Theorem

Which number cannot be expressed as the sum of three squares?

• 5

• 6

• 7

• 8

Fermat's Christmas Theorem states that:

• Every prime of the form 4k + 1 can be expressed as the sum of two squares.

• Every integer can be expressed as the sum of four perfect squares.

• Every even integer greater than 2 can be expressed as the sum of two prime numbers.

• Every integer greater than 2 can be expressed as the sum of two prime numbers.

Which of the following statements is true?

• Every integer can be expressed as the sum of three perfect squares.

• Every integer can be expressed as the sum of two perfect squares.

• Every integer can be expressed as the sum of a square and twice another square.

• Every integer can be expressed as the sum of three prime numbers.

What does Minkowski's Theorem state?

• Every even integer greater than 2 can be expressed as the sum of two prime numbers.

• Every integer can be expressed as the sum of four perfect squares.

• Every convex body in n-dimensional space with vol > 2ⁿ contains at least one lattice point.

• Every integer greater than 2 can be expressed as the sum of two prime numbers.

What does Lagrange's Four-Square Theorem state?

• Every positive integer can be expressed as the sum of four perfect squares.

• Every integer greater than 2 can be expressed as the sum of two prime numbers.

• Every positive integer can be expressed as the sum of three prime numbers.

• Every prime of the form 4k + 1 can be expressed as the sum of two squares.

How could you verify Fermat's Christmas Theorem for a specific prime number?

• Check if the prime number is congruent to 1 modulo 4.

• Try to express the prime number as the sum of two squares.

• Check if the prime number is congruent to 3 modulo 4.

• Divide the prime number by 4 and check the remainder.

Which theorem generalizes Fermat's Christmas Theorem to represent integers, not just primes, as sums of squares?

• Euler's Theorem

• Wilson's Theorem

• Fermat's Last Theorem

• Fermat's Two-Square Theorem

Which of the following numbers can be represented as the sum of two squares?

• 14

• 23

• 25

• 33