Minimizing the Number of Carries in Addition

When numbers are added in base b in the usual way, carries occur. If two random, independent 1-digit numbers are added, then the probability of a carry is b−1 2b . Other choices of digits lead to less carries. In particular, if for odd b we use the digits {−(b− 1)/2,−(b− 3)/2, . . . , . . . (b− 1)/2} then the probability of carry is only b −1 4b2 . Diaconis, Shao and Soundararajan conjectured that this is the best choice of digits, and proved that this is asymptotically the case when b = p is a large prime. In this note we prove this conjecture for all odd primes p. 1 The problem and result When numbers are added in base b in the usual way, carries occur. If two added one-digit numbers are random and independent, then the probability of a carry is b−1 2b . Other choices of digits lead to less carries. In particular, if for odd b we use balanced digits, that is, the digits {−(b− 1)/2,−(b− 3)/2, . . . , 0, 1, 2, . . . (b− 1)/2} then the probability of carry is only b 2−1 4b2 . Diaconis, Shao and Soundararajan [4] conjectured that this is the best choice of digits, and proved that this conjecture is asymptotically correct when b = p is a large prime. More precisely, they proved the following. Theorem 1.1 ([4]) For every > 0 there exists a number p0 = p0( ) so that for any prime p > p0 the probability of carry when adding two random independent one-digit numbers using any fixed set of digits in base p is at least 14 − . The estimate given in [4] to p0 = p0( ) is a tower function of 1/ . Here we establish a tight result for any prime p, proving the conjecture for any prime. Theorem 1.2 For any odd prime p, the probability of carry when adding two random independent one-digit numbers using any fixed set of digits in base p is at least p 2−1 4p2 . ∗Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. Email: [email protected]. Research supported in part by an ERC Advanced grant, by a USA-Israeli BSF grant and by the Israeli I-Core program.