Page:Indian mathematics, Kaye (1915).djvu/28

(14)${\displaystyle ax+1=s^{2},\quad bx+a=t^{2}}$
(15)${\displaystyle 2(x^{2}-y^{2})+3=s^{2},\quad 3(x^{2}-y^{2})+3=t^{2}}$
(16)${\displaystyle ax^{2}+by^{2}=s^{2},\quad ax^{2}-by^{2}+1=t^{2}}$
(17)${\displaystyle x^{2}+y^{2}\pm 1=s^{2},\quad x^{2}-y^{2}\pm 1=t^{2}}$
(18)${\displaystyle x^{2}-a\equiv x^{2}-b\equiv o~Mod.~c}$
(19)${\displaystyle ax^{2}+b\equiv o~Mod.~c}$
(20)${\displaystyle x+y=s^{2},\quad x-y=t^{2},\quad xy=u^{3}}$
(21)${\displaystyle x^{3}+y^{3}=s^{2},\quad x^{2}+y^{2}=t^{3}}$
(22)${\displaystyle x-y=s^{3},\quad x^{2}+y^{2}=t^{3}}$
(23)${\displaystyle x+y=s^{2},\quad x^{3}+y^{2}=t^{2}}$
(24)${\displaystyle \scriptstyle {x^{3}+y^{2}+xy=s^{2},\quad (x+y)s+1=t^{2}}}$
(25)${\displaystyle \scriptstyle {ax+1=s^{3},\quad as^{2}+1=t^{2}}}$
(26)${\displaystyle \scriptstyle {wxyz=a(w+x+y+z)}}$
(27)${\displaystyle \scriptstyle {x^{3}-a\equiv o~Mod.~b}}$
(28)${\displaystyle {\begin{aligned}\\&\scriptstyle {x+y+3=s^{2},~x-y+3=t^{2},~x^{2}+y^{2}-4=u^{2},}\\&\scriptstyle {x^{2}-y^{2}+12=v^{2},\ {\frac {xy}{2}}+y=w^{3},}\\&\scriptstyle {s+t+u+v+w+2=z^{2}}\\\end{aligned}}}$
(29)${\displaystyle \scriptstyle {{\frac {\sqrt[{3}]{xy+y}}{2}}+{\sqrt {x^{2}+y^{2}}}+{\sqrt {x+y+2}}+{\sqrt {x-y+2}}+{\sqrt {x^{2}-y^{2}+8}}=t^{2}}}$
(30)${\displaystyle {\begin{aligned}\\&\scriptstyle {w+2=a^{2},~x+2=b^{2},~y+2=c^{2},~z+2=d^{2},}\\&\scriptstyle {wx+18=e^{2},\ xy+18=f^{2},\ yz+18=g^{2},}\\&\scriptstyle {a+b+c+d+e+f+g+11=13}\\\end{aligned}}}$.

14. Rational right-angled triangles.—The Indian mathematicians of this period seem to have been particularly attracted by the problem of the rational right-angled triangle and give a number of rules for obtaining integral solutions. The following summary of the various rules relating to this problem shows the position of the Indians fairly well.