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Public key 3, 4 fits the equation of y^2 = x^2 + 7

Yet this comes up as an "invalid public key". How is that invalid? Sure, it deviates extremely from other numbers, but what other equations are being applied to the public key to determine its validity?

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    The formula is y^2 = x^3 + 7. I don't think there are any small-number solutions.
    – Pieter Wuille
    Jul 13, 2014 at 18:45
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    Not sure this is what you are talking about, but apparently y^2=x^3+7 has no integral solutions.
    – Murch ♦
    Jul 14, 2014 at 15:05
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    That's OK because secp256k1 doesn't just use plain integers, the equation above is evaluated with integers modulo a very large prime N. Really the equation is: y^2 = x^3 + 7 mod N.
    – David Grayson
    Feb 2, 2015 at 18:46

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You got the formula wrong.

The correct formula is

y2 = x3 + 7 mod N

where N is a very large prime.

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