The first extensive number-theoretical computation run on the world's first U.S. digital general-purpose electronic computer, the ENIAC, is reconstructed. The problem, computing the exponent of 2 modulo a prime, was set up on the ENIAC during a week-end in July 1946 by the number-theorist D.H. Lehmer, with help from his wife Emma and John Mauchly. Important aspects of the ENIAC's design are presented and a reconstruction of the implementation of the problem on the ENIAC is discussed in its salient points. Key words: ENIAC, Derrick H. Lehmer, number theory, Fermat's little theorem, early programming, parallelism, prime sieve