[Shirsendu Roy]

Let n has an odd number of divisors

then this has 1+prime factors each to even power.

If n consists of 1+prime factors each to an even power

it must be a perfect square.

[Shirsendu Roy]

Let real constant t=sqrt{sqrt{8}-2}

so that frac{t^{4}}{4}+t^{2}=1

then (-x^{2}+frac{t^{2}}{2}y^{2})^{2}+(ty^{2}+x)^{2}+(txy-y)^{2}=x^{4}+y^{4}+x^{2}+y^{2}.

[Shirsendu Roy]

13

p congruent to 3(mod 4)

given 4|(p-3)

or, p-3=4k

or, p=4k+3 k is in set of integers

or, p-1=4k+2=2(2k+1) which is given equation

by fermat's theorem

2^{p-1}4^{p-1}6^{p-1}....(p-1)^{p-1}is congruent to 1(mod p)

or, {(2)(4)(6)....(p-1)}^{p-1}is congruent to 1(modp)

or,{(2)(4)(6).....(p-1)} is congruent to 1^{\frac{1}{p-1}}(modp)

or, {(2)(4)(6)....(p-1)} is congruent to 1^{\frac{1}{2(2k+1)}(mod p)

or, {(2)(4)(6)....(p-1)} is congruent to =+-1(modp)

[Shirsendu Roy]

A(0,0),B(a,0) C(x,y)

tanAtanB/2=2 is a given equation

2tan(B/2)/{1-tan^{2}(B/2)}=tanB

or, 2{2/(y/x)}/[1-{2/(y/x)}^{2}]=(y-0)/(x-a) putting given equation where tanA=y/x

or, 4xy/(y^{2}-4x^{2})=(y-0)/(x-a)

or, y^2=4x(2x-a) which is required equation of parabola.

[Shirsendu Roy]

corrected answer given of the same question.

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