@amWhy @XanderHenderson: This is really not math. This user has been posting a ton of "is this true?" questions most of which are like this. Given the absolutely zero context, I would guess that this user just factorized the difference between the two prime factors given, and out popped 59...
@EnzoCreti: How many times are you planning to change your question? Everytime user Henry comes us with a counter-example, you change your condtions. — Tito Piezas IIIFeb 22 at 11:22
@user21820 #14 would be a nice problem if you allowed infinite number of students and problems but asked whether there must be at least one problem solved by infinitely many students.:)
@user21820 right that's why you have to change that bit. Either to finitely many or just one. You can't use half reasonably. You could ask if there must be infinitely many problems solved by infinitely many students but that's too trivial to show. limiting to 1 works better.
@user21820 Assuming we have a countably infinite class of students and a test with countably infinitely many questions and given that every student solved infinitely many questions, must it follow that at least one problem was solved by infinitely many students?
@user21820 Hmm I think you're assuming AC at some point. I wonder where. Countably many countable sets can be uncountable without AC. So I suppose yeah if you label them to begin with you're good. But I wonder if there must be a labeling.
The correct generalization would be: A class has students S and a test has questions Q, and for each x∈S let f(x)⊆Q be the questions that x solved. If for every x∈S there is no injection from f(x) into Q∖f(x), must there exist y∈Q such that there is no injection from { x : x∈S ∧ y∈f(x) } into { x : x∈S ∧ y∉f(x) }?
This is the Realm of Simply Beautiful…
This is the Realm of Simply Beautiful…