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Of particular significance: since the gap never gets always-bigger than 70 million, and there are an infinite number of gaps between primes, by something resem
by lotharbot 13y ago
Of particular significance:
since the gap never gets always-bigger than 70 million, and there are an infinite number of gaps between primes, by something resembling the pigeonhole principle it can be shown that there are an infinite number of prime pairs for at least one specific gap size between 2 and 70 million. That is, there might not necessarily be an infinite number of pairs of primes exactly 2 apart, but there are an infinite number of primes exactly N apart for some N<70,000,000.
- CurtMonash 13y agoThe "something resembling the pigeonhole principle" is the fact that if the union of finitely many sets is infinite, then at least one of the sets must be infinite as well. I don't know whether it has a name.
- jb17 13y agoI think it's usually just called the pigeonhole principle. It's a special case of the Infinite Ramsey Theorem.
- CurtMonash 13y agoPerhaps. But it's simpler than all that. Assume that 70 million disjoint sets all have finite cardinality. Then the cardinality of the union is the sum of the cardinalities, and is finite by construction. Contradiction; QED.