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Answer 1

Is Euler Gamma is Rational ?

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The Euler-Mascheroni constant gamma, sometimes also called 'Euler's constant' or 'the Euler constant' (but not to be confused with the constant e=2.718281...) is defined as the limit of the sequence

$\small{ \gamma = \lim_{(n \mapsto \infty)}( \sum_{(k=1)}^{(n)} \frac1k- \ln \: n) . \dots..}$

Decimals Represented as :-)

• 0.5772156649015328606065120900824024310421593359399235988057672348...

Continued Fraction ):-

• [0; 1, 1, 2, 1, 2, 1, 4, 3, 13, 5, 1, 1, 8, 1, 2, 4, 1, 1, 40, 1, 11, 3, 7, 1, 7, 1, 1, 5, 1, 49, 4, ...]

$\small{ \gamma = \sum_{(k=2)}^∞{ (ζ(k) (-1)^k)/k}}$

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Answer 2

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The number γ has not been proved algebraic or transcendental.

In fact, it is not even known whether γ is irrational.

Using a continued fraction analysis, Papanikolaou showed in 1997 that if γ is rational, its denominator must be greater than 10244663.