The number of solutions of the equation x ^2+ | vedclass

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Question

(a)

Any prime other than 2,3 is of the form $6 k \pm 1$

$	herefore 6 k \pm 1= p$

$6 \lambda+1= p ^2$

$(I)$ If $2 \& 3$ are not soluti

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Vedclass · Answer

(a)

Any prime other than 2,3 is of the form $6 k \pm 1$

$	herefore 6 k \pm 1= p$

$6 \lambda+1= p ^2$

$(I)$ If $2 \& 3$ are not solutions

$x^2 \equiv 1(\bmod 6)$

$y ^2 \equiv 1(\bmod 6)$

LHS $\equiv 2(\bmod 6)$

$RHS \equiv 3(\bmod 6)$

$(II)$ If $2$ is not then all are odd

$LHS$ $\equiv$ Even

$RHS$ $\equiv$ Odd

$C - I$ when $x = y =2$

$a^2+b^2+2=8$

Let $a =2$ then $b ^2+ c ^2=4$

which is not possible for prime ' $b$ ' \& 'c'

$C-II$ $x =2 ; y 
eq 2$

$4+ y ^2= a ^2+ b ^2+ c ^2$

Not possible

$C-III$ $x 
eq 2 ; y 
eq 2$

Not possible

The number of solutions of the equation $x ^2+ y ^2= a ^2+ b ^2+ c ^2$. where $x , y , a , b , c$ are all prime numbers, is

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