-0.46716465011

This number is neither prime nor composite.

The Alternating Prime-Gap Constant is the real number defined by [P=sum_{n=1}^{infty}frac{(-1)^n(p_{n+1}-p_n)}{n^2}], where (p_n) denotes the n-th prime and (p_{n+1}-p_n) is the n-th prime gap. This constant provides a weighted alternating aggregate of consecutive prime gaps. The quadratic denominator ensures absolute convergence, while the alternating signs emphasize oscillatory behavior in the sequence of prime gaps. It defines a new numerical invariant associated with the distribution of consecutive primes. The associated Dirichlet-type generating function is [F(s)=sum_{n=1}^{infty}frac{(-1)^n(p_{n+1}-p_n)}{n^s}], with (P=F(2)). Using unconditional bounds on prime gaps, the defining series converges absolutely. Numerical computation gives [Papprox -0.46716465011.] The arithmetic nature of the constant (for example, whether it is rational, irrational, or transcendental) is currently unknown. [Gautam]