Software 239
Algorithms
Back in the 1930s no mechanical system could perform a calculation with any speed. People still used pencil and paper for most things; the newly-invented mechanical cash registers were slow and could perform only one calculation for each crank of the handle. If you wanted to calculate something complex, you had to employ a computer: a person who could do mental arithmetic enormously fast. Richard Feynman’s first job was computing for the Manhattan Project. The question was: Could a computer, either mechanical or human, blindly follow known rules to decide all mathematical questions? Hilbert’s 10" Problem asked this question of a particular type of mathematical expression — called a Diophantine equation.
Hilbert’s 10" Problem
“Given a Diophantine equation with any number of unknown quantities, devise a finite process to determine whether the equation is solvable in rational integers.”
David Hilbert
Diophantus lived in ancient Persia — now Iran. His son died young and Diophantus was so consumed by grief he retreated into mathematics. He left us seven books of mathematical puzzles — some he devised himself and some of them taken from antiquity. The puzzles look deceptively simple and are all based on equations using whole numbers. His most famous puzzle is set in a poem which tells how old Diophantus was when he died. Can you solve it?
“Here lies Diophantus,; the wonder behold. Through art algebraic, the stone tells how old: ‘God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage, after attaining half the measure of his father’s age, life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.”
Diophantine puzzles look straightforward. Hilbert asked if these
problems could be solved by a mechanical procedure, in modern terms, by an algorithm. To show you what is meant by this, allow me to take you
HOUSE_OVERSIGHT_015929