By the "290 theorem," integral instances of these expressions take on all integer values if they take on all positive integer values up to 290. For 10 points each:
[10m] Name these multivariate polynomials whose terms are all of degree two. These polynomials can be written in the form, "x-transpose A x" for some symmetric matrix A.
ANSWER: quadratic forms [prompt on forms; reject “quadratics”]
[10e] This mathematician proved that all primes of the form “x squared plus y squared” are equivalent to 1 mod 4. Another theorem of this mathematician states that “a to the n plus b to the n equals c to the n” only has integer solutions for n equals 2.
ANSWER: Pierre de Fermat [accept Fermat's last theorem]
[10h] The number of equivalence classes of binary quadratic forms with discriminant d is equal to this quantity for the quadratic field "Q adjoin root d." There are nine quadratic fields for which this quantity is one by the Stark-Heegner theorem.
ANSWER: class number
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