Lowest Integers Representing Sides of a Right Triangle

Leonard E. Dickson
1894 The American mathematical monthly  
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more » ... out Early Journal Content at http://about.jstor.org/participate--jstor/individuals/early-journal--content. JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not--for--profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. cision was made, it was unalterable. He never made an open profession of religion, yet he was an Intensely religious man. He rested his hopes ou the sacrifice of the tender and loving Saviour, and we feel satisfled that he has entered into that rest which remaineth for the people of God. Professor Seitz was not only a mathematician, but he was eminently proficient in other branches of knowledge. His mind was cast in a gigantic mold, "Being devout in heart as well as great in intellect, 'signs and quantities were to him but symbols of God's eternal truth' and he 'looked up through nature up to natures God.' Professor Seitz, in the very appropriate words of Dr. Peabody, regarding Benjamin Peirce, Professor of Mathematics and Astronomy in Harvard University, 'saw thingus preciselv as they are seen by the infinite mind. He held the scales and compasses with which the eternal wisdom built the earth and meted out the heavens. As a mathematician, he was adored with awe. As a mian, he was a christian in the whole aim and tenor of life.' :No mathematician was so universally loved and honored by his contemporaries as was Professor Seitz. He did not gain his knowledge from books, for his librarv consisted of only a few books and periodicals. He gained such a profound insight into the subtle relation of numbers by close application with wlhich he was particularly gifted. He was not a mathematical genius, that is, as usually understood, one who is born with mathematical powers fully developed; but -he was a genius in that he was especially gifted with the power to concentrate his mind upon any subject he wished to investigate. This happy faculty of concentrating all his powers of mind upon one topic to the exclusion of all others and viewing that topic from all sides, enabled him to proceed with certainty. Thread by thread and step by step, he took up,and followed oftt,long lines of thought and arrived at correct conclusions. The darker and more subtle the question appeared to the average mind, the more eagerly he investigated it. No conditions were so complicated as to discourage him. He left a wife and four sons, one of whom has gone to join his father In ths realms of eternal peace. His mother, now (1894) eighty-six years old is still living and enjoying good healthi. ., Fellow in Pure Mathematics, University of Texas. Let the whole numbers expressing the lengths of the sides of a rightangled triangle be reduced to their lowest forms by dividing out their highest common divisor. Call the resulting numbers a, b, and c. 1. They can not all be eiven numbers. For if so, they would still have the common divisor 2. 2. They can not all be odd numbers. For a2 +b2 -c2; and, if a and b are odd, there squares are odd, and the sum of their squares even. But C!2 being even, c must be even.
doi:10.2307/2968644 fatcat:v26fyb6fnbbu3byk7nazlnww2e