设a,b,c是不为1的整数,(a,b,c)=1,且1/a+1/b=1/c,求证:a+b,a-c,b-

1/a + 1/b = 1/c，ab/( a + b ) = c；ab/( a + b )= c( c + 1 )^2/( c + 1 )^2= [ c( c + 1 ) * ( c + 1 ) ] / [ c( c + 1 ) + ( c + 1 ]即a = c( c + 1 )，b = c + 1时，1/a + 1/b = 1/c成立；譬如，c = 2，a = 2( 2 + 1 ) = 6，b = 2 + 1 = 3，1/6 + 1/3 = 1/2；再如，c = 3，a = 3( 3 + 1 ) = 12，b = 3 + 1 = 4，1/12 + 1/4 = 1/3；等等。此时，a + b =c( c + 1 ) + ( c + 1 ) = ( c + 1 )^2；a - c =c( c + 1 ) - c = c^2；b - c = c + 1 - c = 1；即a + b ，a - c ，b - c 都是完全平方数。证毕。