TY - JOUR
T1 - Minimizing the number of carries in addition
AU - Alon, Noga
PY - 2013
Y1 - 2013
N2 - 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 b2-1/4b2. 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.
AB - 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 b2-1/4b2. 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.
KW - Carry
KW - Modular addition
UR - http://www.scopus.com/inward/record.url?scp=84876932372&partnerID=8YFLogxK
U2 - 10.1137/120890612
DO - 10.1137/120890612
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AN - SCOPUS:84876932372
VL - 27
SP - 562
EP - 566
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
SN - 0895-4801
IS - 1
ER -