document.write( "Question 1100513: Please help me to solve the following problem.
\n" ); document.write( "The following equation must be shown to have no solution: 2x^4+2y^4=z^4\r
\n" ); document.write( "\n" ); document.write( "I have come to realize that this equation is a diophantosian equation and there is really no solution to the set of integers.
\n" ); document.write( "My teacher told me that the Euler's theorem should be used for the solution, but I can not do it. \n" ); document.write( "

Algebra.Com's Answer #715242 by ikleyn(52787) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( " I will prove it in 3 (three) steps.\r
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document.write( "    STATEMENT:\r\n" );
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document.write( "        If   +  =   and x, y and z are integer numbers, then both  x  and  y are even numbers.\r\n" );
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document.write( "PROOF of the statement.\r\n" );
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document.write( "For the proof, present  x = 8m + i,  y = 8n + j,  where 0 <= i <=7,  0 <= j <= 7.\r\n" );
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document.write( "Then write\r\n" );
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document.write( "     +  =  +  = (use the binomial formula) = (sum of the terms multiple of 8) +  + .\r\n" );
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document.write( "Below I prepared a rectangular table containing  the numbers   mod 8  with two entries  and \r\n" );
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document.write( "The table has two entries columns at the left and two entries rows at the top:\r\n" );
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document.write( "		1	2	3	4	5	6	7	<<< j\r\n" );
document.write( "i   i^4 mod 8	1	0	1	0	1	0	1	<<< j^4 mod 8\r\n" );
document.write( "1	1	2	1	2	1	2	1	2	\r\n" );
document.write( "2	0	1	0	1	0	1	0	1	\r\n" );
document.write( "3	1	2	1	2	1	2	1	2	\r\n" );
document.write( "4	0	1	0	1	0	1	0	1	\r\n" );
document.write( "5	1	2	1	2	1	2	1	2	\r\n" );
document.write( "6	0	1	0	1	0	1	0	1	\r\n" );
document.write( "7	1	2	1	2	1	2	1	2\r\n" );
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document.write( "The most left column simply lists the numbers  (remainders)  i = 1, 2, 3, 4, 5 , 6, 7 (mod 8).\r\n" );
document.write( "The next column contains the number (remainders)   mod 8.\r\n" );
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document.write( "The most upper row lists the numbers  (remainders)  j = 1, 2, 3, 4, 5 , 6, 7 (mod 8).\r\n" );
document.write( "The second row contains the number (remainders)   mod 8.\r\n" );
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document.write( "The table itself contains, as I just said, the sums   mod 8.\r\n" );
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document.write( "From the table you can see that   is multiple of 8 if and only if BOTH i and j are EVEN numbers. \r\n" );
document.write( "                                                                    (Then and only then we have 0 (zero) in the Table).\r\n" );
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document.write( "Thus the lemma is proved.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Step 2. (To warm your mind)\r
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document.write( "Let the numbers x, y and z are the solution in integer numbers to the given equation:\r\n" );
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document.write( " + = .\r\n" );
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document.write( "Then the left side is an even number.\r\n" );
document.write( "Hence, right side is an even number.\r\n" );
document.write( "Then   is an even number.\r\n" );
document.write( "It implies that z itself is an even number:  z = , where  is integer.\r\n" );
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document.write( "Then the given equation can be written in the form\r\n" );
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document.write( " +  = .\r\n" );
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document.write( "Reduce/(cancel) the factor 2 in both sides.  You will get\r\n" );
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document.write( " +  = .\r\n" );
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document.write( "Now, according to the lemma,  both x and y are even numbers.\r\n" );
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document.write( "This chain of arguments opens the way for the \"infinite descent\" method.\r\n" );
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document.write( "     (The method is attributed to Pierre Fermat and was used by Leonard Euler).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Step 3. (Formal proof)\r
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document.write( "Let assume that equation \r\n" );
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document.write( " + = \r\n" );
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document.write( "has the solution in integer numbers and the triple (x,y,z) IS the solution.\r\n" );
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document.write( "It is clear that if all three numbers x, y and z are even, we can cancel all the three numbers by this common divisor 2, \r\n" );
document.write( "and the new triple also will be the solution to the same equation.\r\n" );
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document.write( "We will cancel this common divisor 2 as many times as possible.\r\n" );
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document.write( "Finally, we can assume that we got (we have) the triple, in which NO MORE THAN ONE member of \"x\" and \"y\" is multiple of 2.\r\n" );
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document.write( "     (If BOTH \"x\" AND \"y\" ARE multiples of 2, then obviously the third number z is also multiple of 2 - the case which we just EXCLUDED).\r\n" );
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document.write( "OK. So, let us assume first that no one of the three terms x, y, z is multiple of 2.\r\n" );
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document.write( "But the LEMMA IMPLIES that BOTH x and y are even numbers.\r\n" );
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document.write( "          CONTRADICTION.\r\n" );
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document.write( "OK. So, let us assume next that only one of the two terms  \"x\"  and  \"y\"  is multiple of 2.\r\n" );
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document.write( "But the LEMMA IMPLIES again that BOTH x and y are even numbers.\r\n" );
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document.write( "          CONTRADICTION.\r\n" );
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\n" ); document.write( "\n" ); document.write( "This contradiction proves that our starting assumption that the given equation has the solutions in integer numbers was wrong.\r
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\n" ); document.write( "\n" ); document.write( " The proof is completed and the problem is solved.\r
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