Indefinite equation is one of the important contents of number theory,In this paper,the solution of two kinds of typical high order indefinite equations was discussed.First,by using the property of division,the solution of the indefinite equation,etc.,the positive integer solution of Mx(x+1)(x+2)(x+3)=Ny(y+1)(y+2)(y+3),was studied at the condition of M=52k,N=1,it is proved that there is no positive integer solution to this indefinite equation.Secondly,by using elementary methods such as recursive sequences,congruences,properties of solutions to the Pell equation,and Legendre symbols,it has been proven that an indefinite equation x3-1=1547y2 only has trivial integer solutions(1,0)the indefinite equation x3+1=1547y2 only has trivial integer solutions(-1,0)when D=1547 for the indefinite equation x3±1=Dy2.
关键词
不定方程/正整数解/整除/同余/Pell方程/勒让德符号
Key words
indefinite equation/positive integer solution/integer division/congruence/pell equation/Legendre symbol
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