Until I get more creative, here's something interesting.

It is a comment I made here: Cicadas: ready for prime time. At the time of publishing this post it is still awaiting approval.

The article talks about a particular variety of cicadas who show up only every 13 year and are currently screeching/chirping around in Alabama and Arkansas (USA states). The author explains one existing theory why they show up after such a strange number of years. I just added my two pennies worth. Here's my nerdy comment. Have fun ;)!

*"Cool explanation. I thought a little about it, thought I would present the thoughts to maybe make this 'more complete' or at least add food for thought.*

*I think LCM (Lowest Common Multiple, since different people learnt different names for it in school) has a very important part in this, even more than a*

*prime. The LCM of two numbers is their product if they have no factors in common.*

*e.g.*

*6 & 7 - no factors common. LCM is 42 (6X7)*

*6 & 9 - 3 is common factor, so LCM = 18 (6X3 or 9X2) is smaller than above even though 9 is larger than 7*

*So if our 13 year friend has 39 year cycle predator, every generation of the predator would still get a nice cicada buffet, even though that would be every*

*third cicada generation.*

*On other hand, if our cicada had a 10 (2*5) year cycle, which is definitely not a prime, the twain shall meet every (2 * 5 * 3 * 13) ie 390 years!!! that way*

*38 cicada generations would heave a sigh of relief, and every 39th would be screwed. But that 'years of peace' is just the direct advantage. Additinally, the*

*same predator's 9 generations would be devoid of cicada. If cicada were vital to their survival, they might be very dwindled in numbers, if not gone totally*

*extinct. So more reasons for the 39th cicada generation to rejoice!*

*--->So, if the cicada is looking at saving itself from just ONE particular predator ( as your article says "these loud insects are trying to evade A*

*predator"), it would get maximum advantage by 1) having a life cycle years number which has no factors common with predator life cycle years number, even if*

*it is not a prime. 2) Within constraints of condition 1, having as large cycle as possible.*

*But, therein comes the statistical aspect, and an important one. Does the cicada have to worry about just one predator? Most likely, no. And even if yes,*

*does the cicada/nature know what the life cycle of this predator is? Again, we could lean in favour of no. In such a 'blind' or 'random' case, cicada's best*

*option is to go for a prime number, because it can only have factors common with predator if the predator's cycle is a multiple of it's cycle. I that case,*

*the twain shall meet at every preadtor generation (tough luck!) for such special 'multilple of cicada year's predator, and will meet at large intervals*

*(product of cicada & predator's cycle years) for every other predator!!!*

*--->So yeah, that way primes are best, if we don't know the predator's nature!*

*And hopefully, the things the cicadas eat do grow every year, otherwise such prime cicada's would hit puberty, come out happy to find no predators, and die*

*hungry!!*

*Also, this theory makes sense only if, the 'adult cicada' eating preadtors themselves are in a biological state of feeding on cicadas for only 1 year (or*

*such short duration of time that the cicadas are out)*

*Otherwise, say the cicada has 13 year cycle, and predator has 30 year cycle. Mathematician will predict many happy care-free generations for the cicada, (13*

** 30 = 390, so only 390th generation needs to worry). But suppose out of those 30 years, the predator is in a state that can feed on cicadas for 5 years*

*(adulthood suppose). Then well, the cicada and this theory are both screwed. Like this...*

*say year 0 of the cicada (13th year of last generation) was year 0 of predator (25th year of last generation). Then in the 13th year 1st generation cicadas*

*are safe. But in the very next generation ie. in the 26th year, the cicadas are out just when the predator starts feeding! Shit! So if the predator has a*

*large 'feeding band' of years, prime or its neighbour non prime would not make heck of a difference. cicadas with 13, 14 or 15 years cycle would be toast in*

*their second generation. Rather the cicadas with 12 year cycle (non prime) will be safe at least in second generation. This is just a knee jerk thought, and*

*a very conveniently picked example to illustrate it. I have yet to think whether primes still cope better with such 'band feeding duration' predators in the*

*long run. I think in this case the optimum number might be a function of the 'ideal cicada prime' (if predator was not banded duration feeded), the predator*

*cycle, and the predator feeding band years. Just an intuition. Maybe will work on it later, maybe not."*

## 2 comments :

Glad for the deeper analysis! Thanks. This further thinking you've put into this is exactly what I hoped to do with my column. Your LCM reasoning leads to the conclusion that the two life-cycles need to be relatively prime.

This is the looooooooooooongeest post i have read (read as seen) of yours.

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