Unique phone numbers in India

When I posted mine and other teaching assistant’s mobile numbers on IITB Moodle and mentioned that my number has two prime factor, in case you wants to verify; and others should be right since I copy and pasted them, Srihari  replied back,

Sunny got 3 prime factors and you 2 as you said..while achin had 6..
now i started to wonder how many prime factors on avg for 10 digit telephone number (starting with 9).. any clues?
sorry for spamming.

Well, unfortunately these kind of emails are considered spam in Indian Universities. But anyway, here is what I am able to reply,

Achin number is then probably worth envy. One can do the simulation. But theorem proving will be good. The maximal is 2x3x5x7x11x13x19x23x29 = 6469693230. It has ten distinct prime factors. But it does not start with 9. And another multiplication will make it 11 digit. If we drop one of the prime in this list, we can check whether there exists another prime which can be used as replacement. In fact, 41 is a prime which can do that. Drop 29, and multiply by 41 we’ll get 9146807670 and 9592993410 with 43. What about other numbers with 10 prime factors? If we drop two primes e.g. 23 and 29, then product of rest becomes 9699690. Then we need two distinct prime such that their product is somewhere between 928 and 1030 (else the total product will not have 9 as its first digit). There aren’t any! SO THERE ARE ONLY TWO NUMBERS WITH TEN PRIME FACTORS. I called them up. First one is in Maharashtra (it was switched off) and second one is in Haryana (He did not know what the hell is a prime number so he gave the phone to his brother, he also did not know and I did not explain.)

One can find many numbers with 9 prime factors by dropping 23 and multiplying by others (this makes 223092870 one of the mothers of all such numbers). This mother have, in addition to the above two, following children,

9369900540
9592993410
9816086280

One sure can take clues from it. I have dropped 29 and multiplied with other numbers (41, 42, etc). You can drop others and figure out. They should not be more than 4 children for each mother (proof?). I can establish upper bound by 36 (surely less than that). Now since the problem is yours, you deserve the joy of solving it fully. This was a nice one by the way!! smile

So there could be at max 36 such numbers (with at least 9 prime factors, and they are surely less than 36). I wonder who are the lucky bastards possessing them?

REF :
[1] http://primes.utm.edu/howmany.shtml

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Author: Dilawar

Graduate Student at National Center for Biological Sciences, Bangalore.

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