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Every Even Number is the difference of two consecutive Primes in infinitely many ways. If true, taking the difference 2, this conjecture implies that there are infinitely many Twin Primes (Ball and Coxeter 1987). The Conjecture has never been proven true or refuted.
See also Even Number, Twin Primes
References
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed.
New York: Dover, p. 64, 1987.
de Polignac, A. ``Six propositions arithmologiques déduites de crible d'Ératosthène.'' Nouv. Ann. Math. 8, 423-429, 1849.