# Art of problem solving introduction to number theory pdf

If the solution to a problem is easy to check for correctness, is the problem easy to solve? 1,000,000 prize for the first correct solution. For some questions, there is art of problem solving introduction to number theory pdf known way to find an answer quickly, but if one is provided with information showing what the answer is, it is possible to verify the answer quickly. Given a partially filled-in Sudoku grid, of any size, is there at least one legal solution?

Thousands of other problems seem similar, fast to check but slow to solve. Decades of searching have not yielded a fast solution to any of these problems, so most scientists suspect that none of these problems can be solved quickly. However, this has never been proven. Sudoku, can also be solved in polynomial time. 1971, there were previous inklings of the problems involved, the difficulty of proof, and the potential consequences. In 1955, mathematician John Nash wrote a letter to the NSA, where he speculated that cracking a sufficiently complex code would require time exponential in the length of the key. In such analysis, a model of the computer for which time must be analyzed is required.