{"id":3594,"date":"2023-11-23T07:51:08","date_gmt":"2023-11-23T12:51:08","guid":{"rendered":"https:\/\/sites.williams.edu\/Morgan\/?page_id=3594"},"modified":"2023-11-23T07:52:39","modified_gmt":"2023-11-23T12:52:39","slug":"a-one-page-proof-of-fermat","status":"publish","type":"page","link":"https:\/\/sites.williams.edu\/Morgan\/math-chat-archives\/a-one-page-proof-of-fermat\/","title":{"rendered":"A ONE-PAGE PROOF OF FERMAT?"},"content":{"rendered":"

February 18, 1999<\/p>\n

<\/h3>\n

MATH JOKES.\u00a0<\/b>Math Chat has received a number of mathematical jokes, most of them terrible. This week’s winner is a riddle from Peter Hegarty:<\/p>\n

RIDDLE.\u00a0<\/b>Which mathematical term is named after a well-known American politician?<\/p>\n

The answer is near the end of this column.<\/p>\n

OLD CHALLENGE\u00a0<\/b>(Joe Shipman). Select the best occurrence in the world of each number from 1 to 10. For example, 12 is the number of eggs in a dozen or the number of months in a year.<\/p>\n

ANSWER.\u00a0<\/b>Here is the winning response compiled from John Robertson, Henry Ricardo, Bob Swanson, Jean-Pierre Carmichael, and Ryan Grove:<\/p>\n

 <\/p>\n

    \n
  1. Number of Earth’s moons<\/li>\n
  2. Romeo and Juliet<\/li>\n
  3. Dimensions; Mazur, Ribet, and Wiles (contributors to the proof of Fermat’s Last Theorem)<\/li>\n
  4. The Four Freedoms (FDR, 1941: freedom of speech, freedom of worship, freedom from want, freedom from fear)<\/li>\n
  5. Platonic solids [cube, octahedron, tetrahedron, dodecahedron, icosahedron]<\/li>\n
  6. Legs on insects<\/li>\n
  7. Notes of the musical scale<\/li>\n
  8. Cylinders in a V8 engine (such as in the Ferrari 308 series); vegetables in V8 juice (tomatoes, carrots, celery, beets, parsley, lettuce, watercress, and spinach)<\/li>\n
  9. Planets; Beethoven symphonies<\/li>\n
  10. Fingers<\/li>\n<\/ol>\n

    (Can readers improve on this list?) Then there is this from Al Zimmermann:<\/p>\n

     <\/p>\n

      \n
    1. The amount, in cents, of the recent postage increase for first class US mail.<\/li>\n
    2. The number of sides to every story.<\/li>\n
    3. The number of moving parts in a Wankel engine.<\/li>\n
    4. The number of Beatles.<\/li>\n
    5. The number of players on a basketball team.<\/li>\n
    6. The number of sodas in a six-pack.<\/li>\n
    7. The number of days in a week.<\/li>\n
    8. The number of days in a week, according to the Beatles.<\/li>\n
    9. The number of months in a pregnancy.<\/li>\n
    10. The number of years it seems the Clinton\/Lewinsky scandal has been going on.<\/li>\n<\/ol>\n

      Finally, in his beautiful response, David Shay relates that, “At the end of the Seder night, which begins the Jewish holiday of Passover, it is common to sing a song named ‘Who Knows One.’ This song gives an exact Jewish answer to your challenge, in the range of 1 to 13. Here it is:<\/p>\n

       <\/p>\n

      Thirteen are the attributes of God;<\/p>\n

      Twelve are the tribes of Israel;<\/p>\n

      Eleven were the stars in Joseph’s dream;<\/p>\n

      Ten commandments were given on Sinai;<\/p>\n

      Nine is the number of the holidays;<\/p>\n

      Eight are the days to the service of the covenant;<\/p>\n

      Seven days there are in a week;<\/p>\n

      Six sections the Mishnah has;<\/p>\n

      Five books there are in the Torah;<\/p>\n

      Four is the number of the matriarchs;<\/p>\n

      Three is the number of the patriarchs;<\/p>\n

      Two are the tables of the covenant;<\/p>\n

      One is our God in heaven and earth.”<\/p>\n

       <\/p>\n

      NEW CHALLENGE.\u00a0<\/b>Critique the following short proof of Fermat’s Last Theorem sent in by reader Rob Connelly. (In perhaps the biggest mathematics news of the century, Andrew Wiles recently came up with a very long and complicated proof to this 350-year old problem.)<\/p>\n

      Fermat’s Last Theorem. The equation<\/p>\n

      (1) xn<\/sup>\u00a0+ yn<\/sup>\u00a0= zn<\/sup>\u00a0has no positive integer solutions for n > 2.<\/p>\n

      Proposed proof. Suppose there were such a solution. Since x\u00a0\"not\u00a0y, we may suppose x = y + a, z = y + b, with b > a positive integers. Consider the integer N defined by<\/p>\n

      (2) zn -1<\/sup>\u00a0= xn -1<\/sup>\u00a0+ yn -1<\/sup>\u00a0+ N.<\/p>\n

      Then<\/p>\n

      xn<\/sup>\u00a0+ yn<\/sup>\u00a0= zn<\/sup>\u00a0= z(xn -1<\/sup>\u00a0+ yn -1\u00a0<\/sup>+ N).<\/p>\n

      Solving for N yields:<\/p>\n

      N = [(y+a)\u00a0n -1<\/sup>\u00a0(a-b) + yn -1<\/sup>\u00a0(-b)]\/(y+b) = [F(y)]\/(y+b) ,<\/p>\n

      so y+b divides F(y) and<\/p>\n

      0 = F(-b) = (a-b)\u00a0n\u00a0<\/sup>+ (-b)\u00a0n<\/sup><\/p>\n

      0 = (b-a)\u00a0n\u00a0<\/sup>+ bn<\/sup>\u00a0> 0,<\/p>\n

      the desired contradiction.<\/p>\n

      ANSWER TO RIDDLE.<\/b>\u00a0The mathematical term named after a well-known American politician is “Algorithm.” (Readers are invited to continue to submit more jokes for future columns.)<\/p>\n

      Send answers, comments, and new questions by email to:<\/p>\n

      Frank.Morgan@williams.edu<\/a>, to be eligible for\u00a0Flatland<\/i>\u00a0and other book awards. Winning answers will appear in the next Math Chat. Math Chat appears on the first and third Thursdays of each month. Prof. Morgan’s homepage is at\u00a0www.williams.edu\/Mathematics\/fmorgan<\/a>.<\/p>\n

      Copyright 1999, Frank Morgan.<\/p>\n","protected":false},"excerpt":{"rendered":"

      February 18, 1999 MATH JOKES.\u00a0Math Chat has received a number of mathematical jokes, most of them terrible. This week’s winner is a riddle from Peter Hegarty: RIDDLE.\u00a0Which mathematical term is named after a well-known American politician? The answer is near the end of this column. OLD CHALLENGE\u00a0(Joe Shipman). Select the best occurrence in the world […]<\/p>\n","protected":false},"author":2965,"featured_media":0,"parent":3459,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-3594","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3594","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/2965"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=3594"}],"version-history":[{"count":4,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3594\/revisions"}],"predecessor-version":[{"id":3598,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3594\/revisions\/3598"}],"up":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/pages\/3459"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=3594"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}