Abstract. Fermat's Last Theorem Proved on a Single Page. "The simplest solution is usually the best solution"---albert Einstein

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1 Copyright A. A. Frempog Fermat's Last Theorem Proved o a Sigle Page "5% of the people thik; 0% of the people thik that they thik; ad the other 85% would rather die tha thik."----thomas Ediso "The simplest solutio is usually the best solutio"---albert Eistei Abstrat Hoorable Pierre de Fermat was truthful. He ould have squeezed the proof of his last theorem ito a page margi. Fermat's last theorem has bee proved o a sigle page. The proof is based o the Pythagorea idetity si + os =. Oe will first show that if =. holds, followed by showig that if >, does ot hold. Applyig a polar oordiate system, let ab,,ad be three relatively prime positive itegers whih are the legths of the sides of a right triagle, where is the legth of the hypoteuse, ad a ad b are the legths of the other two sides. Also, let the aute agle betwee the hypoteuse ad the horizotal be deoted by. Three similar versios of the proof are preseted. The proof is very simple, ad eve high shool studets a lear it. Perhaps, the proof i this paper is the proof that Fermat wished there were eough margi for it i his paper.

2 Proof: Versio Pythagorea Idetity Postulate: There exist oly a sigle fudametal trigoometri idetity suh that os. Pla: Oe will first show that if =, holds, followed by showig that if >, does ot hold. Let ab,,ad be three relatively prime positive itegers whih are the legths of the sides of the right triagle i the figure below, where is the legth of the hypoteuse, ad a ad b are the legths of the other two sides. Also, let deote the aute agle betwee the hypoteuse ad the horizotal. The a = os () y b = si () () P = ( os ) + ( si ) (,) = os + si (4) b = (os + si ) (5). Left-had side (LHS) of equatio ( 5 ) equals x right-had side (RHS) of (5) oly if O a os That is, a eessary oditio for (5) to be true is that os If =, = (os + si ), is true sie os + si = ad therefore, equatios (5) ad () hold. Sie there exists oly a sigle Pythagorea idetity suh that os, ad os + si =, with =,, there are o other positive itegers suh that os. Therefore, equatios (5) ad () will be true oly if =, ad there are o other itegers, > whih will make equatios (5) ad () true. Therefore, holds oly if =, ad does ot hold if >. The proof is omplete. Colusio Fermat's last theorem has bee proved i this paper. Note above that the mai riterio is i equatio (5) above, whih requires that os, if = (os + si ) ad are to hold. Perhaps, the proof i this paper is the proof that Fermat wished there were eough margi for it i his paper. Adote

3 a = os b = si = ( os ) + ( si ) = os + si =. (os + si ). Equatio ( 5 ) is true oly if os + si = For (5) to be true os + si =. If =, = (os + si ) is true sie os + si = ad therefore, equatios (5) ad () hold. There exists a sigle idetity suh that os + si =, ad os + si = with =,, there are o other positive itegers suh that os +. si = Therefore, equatios (5) ad () will be true oly if =, ad there are o other itegers, > makig eqs (5) ad () true. = a + b holds oly if =, ad does ot hold if >. QED Proof: Versio i the Margi Fermat was truthful. He ould have squeezed the proof ito the page margi. If Fermat were reiarated, he would be pleased.

4 Proof: Versio Pla: Oe will first show that if =, = a + b holds, followed by showig that if >, does ot hold. Let ab,,ad be three relatively prime positive itegers whih are the legths of the sides of the right triagle i the figure below, where is the legth of the hypoteuse, ad a ad b are the legths of the other two sides. Also, let deote the aute agle betwee the hypoteuse ad the horizotal. The a = os () y b = si () () = ( os ) + ( si ) P (,) = os + si (4) b = (os + si ) (5). Left-had side (LHS) of equatio ( 5 ) equals x right-had side (RHS) of (5) oly if O a os That is, a eessary oditio for (5) to be true is that os If =, = (os + si ), is true sie os + si = ad therefore, equatios (5) ad () hold. If =, (os + si ) sie os + si ad equatios (5) ad () do ot hold. ( os + si = (os + si )(os ossi + si ) Therefore, if =, equatios (5) ad () do ot hold.. If = 4,, (os 4 + si 4 ) sie os 4 + si 4. (os4 + si4 os + si ) { os4 + si 4 = (os + ossi + si )(os ossi + si } Therefore, if = 4, equatios (5) ad (). do ot hold. Replaig by k + or by k + i os, oe obtais respetively A If = k + or B If = k +, os + si = os + + si + os = osk+ + sik+ If osk+ + sik+ = os + si, the osk+ + sik+ = os + si = k + = ad k = ad = + = oly if k =, ad the =. Here also, =. Note: Neessary oditio is i (5) above... Observe that if = k + or = k +, ad oe wats to irease the expoet while at the same time maitaiig that os ( a eessary oditio for (5) ad () to be true), the results show that oe aot do that simultaeously ad that oe should keep the expoet at =. Therefore, equatios (5) ad () will be true oly if =, Therefore, holds oly if =, ad does ot hold if >. Colusio Fermat's last theorem has bee proved i this paper. Note above that the mai riterio is i equatio (5) above, whih requires that os, if = (os + si ) ad are to hold. Perhaps, the proof i this paper is the proof that Fermat wished there were eough margi for it i his paper. Adote 4

5 Proof: Versio Pla: Oe will first show that if =, = a + b holds, followed by showig that if >, does ot hold. Let ab,,ad be three relatively prime positive itegers whih are the legths of the sides of the right triagle i the figure below, where is the legth of the hypoteuse, ad a ad b are the legths of the other two sides. Also, let deote the aute agle betwee the hypoteuse ad the horizotal. The a = os () y b = si () () = ( os ) + ( si ) P (,) = os + si (4) b = (os + si ) (5). Left-had side (LHS) of equatio ( 5 ) equals x right-had side (RHS) of (5) oly if O a os That is, a eessary oditio for (5) to be true is that os If =, = (os + si ), is true sie os + si = ad therefore, equatios (5) ad () hold. os + si = b + a = a + b = a + b a + b = If =, (os + si ) os = a ; os = a si = b ; si = b sie os + si ad equatios (5) ad () do ot hold. os si + = b + a = a + b os + si = (os + si )(os ossi + si ( a+ b)( a ab+ b) = Therefore, if =, equatios (5) ad () do ot hold.. Replaig by k + or by k + i os, oe obtais respetively A If = k + or B If = k +, os + si = os k + + si k + os + si = os k + + si k + If osk+ + sik+ = os + si, the osk+ + sik+ = os + si = k + = ad k = ad = + = oly if k =, ad the =. Here also, =. Note: Neessary oditio is i (5) above... Therefore, equatios (5) ad () will be true oly if =, ad there are o other itegers, > whih will make equatios (5) ad () true. Therefore, holds oly if =, ad does ot hold if >. Colusio Fermat's last theorem has bee proved i this paper. Note above that the mai riterio is i equatio (5) above, whih requires that os, if = (os + si ) ad are to hold. Perhaps, the proof i this paper is the proof that Fermat wished there were eough margi for it i his paper. Adote 5

6 PS Disussio A If = k + C If = k +, os + si = os k + + si k + os + si = os k + + si k + If osk+ + sik+ = os + si, the osk+ + sik+ = os + si = k + = ad k =. = k = oly if k =, ad the =. The egative k value implies that value aot The k value implies that value, = irease if the eessary oditio i (5) is maitaied... aot irease if the eessary The eessary oditio implies that always, =. oditio i (5).is to be maitaied.. The eessary oditio implies that =. B If = k + D If = k +4, os + si = os k si k + 4 os + si = os k + + si k + If osk+ 4 + sik+ 4 = os + si, the k+ k+ os + si = os + si = k + = ad k = 0, ad the =. k value implies that value aot irease beause of the eessary oditio i (5). k + 4 = ad k =. Agai = + 4 = The egative k value implies that value aot irease beause of the eessary oditio i (5). I A, B, C ad D, above, the value remais ostat at =, despite attempt s to irease it. beause of the eessary oditio, os i (5). Note: Apart from os + si =, if there were aother trigoometri idetity suh that os, that value will be aother value satisfyig. Aalogy : If oes feet are o the seod rug of a ladder, ad oe wats to move oe's feet to a higher rug while at the same time maitaiig oe's body o the seod rug, oe will be fored to retur to the seod rug, implyig that oe aot move oes feet to a higher rug while maitaiig oes body o the seod rug. Aalogy : If oe lives o the seod floor of a buildig, ad oe wats to move to a higher floor, say, the third floor, ad the seod floor has a eessary health-are faility, whih is ot available o the higher floors, oe would ot be able to move ad live o the third or a higher floor. Eve if oe tried hard to live o the third floor, oe would be ompelled to move bak to the seod floor beause of the eessary health-are faility o the seod floor. I a fiaial deisio, base your deisio o what is eessary ad ot what you wat. 6