The following results of the thirty-first William Lowell Putnam Mathematical Competition held on December 5, 1970, have been determined in accordance with the regulations governing the Competition. This competition is supported by the William Lowell Putnam Intercollegiate Memorial Fund left by Mrs. Putnam in memory of her husband and is held under the auspices of the Mathematical Association of America. The first prize, five hundred dollars, is awarded to the Department of Mathematics of the University of Chicago, Chicago, Illinois. The members of the team were Robert Israel, Robert A. Oliver, and Robert Tax; to each of these a prize of one hundred dollars is awarded. The second prize, four hundred dollars, is awarded to the Department of Mathematics of Massachusetts Institute of Technology, Cambridge, Massachusetts. The members of the team were David M. Christie, Don Coppersmith, and Steven Winker; to each of these a prize of seventy-five dollars is awarded. The third prize, three hundred dollars, is awarded to the Department of Mathematics of the University of Toronto, Toronto, Ontario, Canada. The members of the team were Daniel Gautreau, Daryl Geller, and Joseph S. Repka; to each of these a prize of fifty dollars is awarded. The fourth prize, two hundred dollars, is awarded to the Department of Mathematics of the Illinois Institute of Technology, Chicago, Illinois. The members of the team were George F. Cornelius, Zbigniew Friedorowicz, and Wayne F. Mroz; to each of these a prize of fifty dollars is awarded. The fifth prize, one hundred dollars, is awarded to the Department of Mathematics of the California Institute of Technology, Pasadena, California. The members of the team were Leonidas Guibas, Andrew M. Odlyzko, and David J. Smith; to each of these a prize of fifty dollars is awarded. The six persons ranking highest in the examination, named in alphabetical order, are Jockum Aniansson, Yale University; Don Coppersmith, Massachusetts Institute of Technology; Jeffrey Lagarias, Massachusetts Institute of Technology; Robert A. Oliver, University of Chicago; Arthur Rubin, Purdue University; and Steven K. Winker, Massachusetts Institute of Technology. Each of these has been designated as a Putnam fellow by the Mathematical Association of America and is awarded a prize of two hundred and fifty dollars. The five persons ranking second highest in the examination, named in alphabetical order, are Daryl Geller, University of Toronto; Zbigniew Friedorowicz, Illinois Institute of Technology; Dale H. Peterson, Yale University; Joseph S. Repka, University of Toronto; and Jonathan Rosenberg, Harvard University. To each of these a prize of one hundred dollars is awarded. The following teams, named in alphabetical order, won honorable mention: University of California at Davis, the members of the team were Dean Hickerson,