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Determine largest number not divisible by sum of multiples of two primes

Posted 8 years ago

I have a simple problem, and wonder if Wolfram has the software to solve it.

The general problem is: find the largest number which cannot be expressed as the sum of nx+my where n and m are integers and x and y are primes. I am looking for the specific case x and y are 7 and 11, but I really want a theoretical approach on how to tackle this problem.

I had earlier posts that were written in a daze and were nonsensical. I apologize to those who wasted their time trying to help me. Hopefully this statement will be both more coherent and a more challenging.

POSTED BY: Julian Svedosh
6 Replies
Posted 8 years ago

You are both quite right ... my question was nonsense as posted. Left out a critical element.

I am seeking the largest number which can be expressed as the sum of products of 7 and 11. Ignore the earlier post. I was having moment of premature senility. Or perhaps just a moment of senility.

Thanks for reading and bringing my sloppiness to my attention.

And I am also interested in the general theory of the distribution of numbers which can be generated as the sums of powers of primes.

POSTED BY: Julian Svedosh

Okay, thanks for the clarification. I'm still confused though. Do you want the largest number not expressible as m*7+n*11 for {m,n}>=0? If so, that would the the Frobenius number. Or are you looking for something else (I ask because this time the phrasing involves "largest number than CAN be expressed as..." instead of "...CANNOT...").

If FrobeniusNumber is in fact what you had in mind, then you might want to look up "Apery sets" to get some possibly relevant links for the more general question (I make no promises as to ultimate relevance though).

POSTED BY: Daniel Lichtblau
Posted 8 years ago

yes , precisely. I know very little about number theory and will need to look up both frobenius numbers and apery sets. Thanks much.

POSTED BY: Julian Svedosh

My advice is look into Frobenius numbers first since that's an easier area. If the primary interest involves just two primes or prime powers, then all the better (since the case of two generators is considerably simpler than the general case). Also if Apery sets gets daunting, a (slightly) simpler term of relevance might be "modular semigroup". Which in this case basically means "sets of integers generated by nonnegative combinations of some given set".

POSTED BY: Daniel Lichtblau

Do you actually mean the largest that cannot be written as a nonnegative integer combination of the two values? That would be the Frobenius number.

FrobeniusNumber[{7, 11}]

(* Out[875]= 59 *)

If this is what you had in mind, then the wording was such that W|A would not have much of a chance, since the phrase "divisible by" is used quite incorrectly. If you meant something else, I defer to the response by Murray Eisenberg.

POSTED BY: Daniel Lichtblau

How do you expect any software to find something that cannot possibly exist?

For every positive integer $n$, the integer $2^n$ is not divisible by both 7 and 11, in fact, not divisible by either. Hence there is no such thing as "the largest number not divisible by both 7 and 11."

POSTED BY: Murray Eisenberg
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