Are there any consecutive integers which don't share letters when spelled out in English (e.g.One, twO)? If not, can you construct a proof?
Firstly consider the early numbers with "irregular" words:
one, Two, Three, Four, Five, Six, Seven, eight, nine, ten, eleven, twelve.
Then the "teens" all share letters, ending with
Then all the twenty-somethings share letters with each other, all the thirty-somethings with each other, etc. The only possible place to get two consecutive numbers not sharing letters is
a multiple of ten and the one just before it. And even that's not possible because they all contain "ty" in the number. Going from Ninety-nine to a huNdred is also fine, and then to a thousaNd, a millioN, a billioN, trillioN, etc.
Even infiNity will be OK :-P