Why Do Periodical Cicadas Emerge on 13- and 17-Year Cycles?

A predator cycling every 2 years waits 26 years to catch a 13-year brood — and 221 years to catch both broods at once. The interval isn’t patience. It’s arithmetic.

Periodical cicadas in the eastern United States emerge on either 13- or 17-year cycles, and both numbers are prime. That overlap is not coincidence. A prime interval shares no common factors with the short cycles — 2, 3, 4 years — that most predator populations run on, so predator booms rarely land on the same year as a cicada emergence.

The math is specific. A predator peaking every 2 years won’t realign with a 13-year brood for 26 years. A predator peaking every 3 years waits 51 years for the 17-year brood. The least common multiple of 13 and 17 themselves is 221 years — meaning both broods only co-emerge on the same calendar year once every two centuries.

Quick Facts
– 13 and 17 are both prime, divisible only by 1 and themselves
– A 2-year predator cycle realigns with a 13-year brood every 26 years
– The LCM of 13 and 17 is 221 years — one shared emergence per two centuries
– Predator-avoidance was formally proposed by Lloyd and Dybas in 1966
– Prime cycles also reduce hybridization between broods, a separate proposed benefit

How Prime-Number Cycles Limit Predator Synchronization

The predator-avoidance hypothesis, first formally proposed by Monte Lloyd and Henry Dybas in 1966 and later popularized by Stephen Jay Gould, rests on a clean mathematical property. A 12-year or 15-year cycle shares common factors with many short predator cycles. A 13-year or 17-year cycle does not. fewer shared factors means fewer overlapping peaks, so predator populations can’t build up across generations timed to a cicada emergence.

When cicadas do surface, they arrive in such volume that individual predation risk collapses — a strategy called predator satiation. The prime interval keeps predator populations from being pre-sized for the event.

Hybridization and the Allee Effect as Additional Mechanisms

Predator avoidance is not the only explanation the biological literature supports. Because 13 and 17 are prime, two broods with different cycles almost never emerge together. That rarity limits hybridization between them. Hybrid offspring at low density suffer higher predation — a dynamic the research literature connects to the Allee effect, where very small populations have reduced fitness.

Modeling work published in PNAS shows that prime-numbered cycles are favored under Allee-effect conditions specifically because rare co-emergence events can be costly enough to select against non-prime intervals. The hybridization and predator-avoidance hypotheses aren’t mutually exclusive; both point toward prime numbers as a durable solution.

Closing

Forty years of mathematical modeling and field observation have not produced a single simpler explanation that fits the data as well. The cicadas don’t calculate anything. The prime intervals persist because individuals on non-prime schedules left fewer descendants, generation after generation, until 13 and 17 were what remained.

Two numbers, each indivisible, still running the same clock they ran before any human thought to call them prime.