query: Proof that every odd integer is a difference of two squares
positive_passages.docid: 18633
positive_passages.text: Prove every odd integer is the difference of two squares
negative_passages.docid: 63240
negative_passages.text: Every two consecutive integers are coprime.
negative_passages.docid: 68092
negative_passages.text: Does square difference prove that 1 = 2?
negative_passages.docid: 134765
negative_passages.text: Is an integer a sum of two rational squares iff it is a sum of two integer squares?
negative_passages.docid: 32350
negative_passages.text: Compilation of proofs for the summation of natural squares and cubes
negative_passages.docid: 123267
negative_passages.text: Proving that even numbers equal the sum of two odd numbers.
negative_passages.docid: 140433
negative_passages.text: The number of ways of writing an integer as a sum of two squares
negative_passages.docid: 41789
negative_passages.text: Fermat's theorem on sums of two squares
negative_passages.docid: 94209
negative_passages.text: Prove by contradiction that every integer greater than 11 is a sum of two composite numbers
negative_passages.docid: 49585
negative_passages.text: Proof a graph is bipartite if and only if it contains no odd cycles
negative_passages.docid: 39156
negative_passages.text: Concerning what is between two consecutive squares.
negative_passages.docid: 94326
negative_passages.text: a square in a finite field of odd order
negative_passages.docid: 142584
negative_passages.text: Confusion in proof that primes $p = 4k + 1$ are uniquely the sums of two squares
negative_passages.docid: 83837
negative_passages.text: Odd/Even Permutations
negative_passages.docid: 128377
negative_passages.text: Mathematical Induction and "the product of odd numbers is odd"
negative_passages.docid: 4061
negative_passages.text: Proving there are infinitely many pairs of square-full consecutive integers
negative_passages.docid: 92166
negative_passages.text: Number of odd binomial coefficients is a power of $2$