Yes, it can be if you’re willing to spend the energy to rip protons out of the nucleus and slam them back in.
Mathematics is absolute. Physics (and the whole physical world) is fuzzy.
Dying Butterfly writes:
> Nothing to contribute except that I read elements as elephants.
Yes, it probably is, but it’s one I’ve seen before, and it makes a certain amount of sense. If you want to know what a number is “made of,” you look at its unique factorization, so that, for example, 28 = 2[sup]2[/sup]7 in the same sense that water = H[sub]2[/sub]O. (One way the analogy breaks down is that, if I understand correctly, there can be more than one way of combining the same elements in the same quantities, with different molecular structures.)
Hypnagogic Jerk is right about primes being fundamental to number theory. So, in some sense, number theory is that “mathematical theory” that the OP asked about.
The other problem with the analogy is that knowing the elemental composition of a molecule can tell you pretty much everything you need to know about it, including how it reacts with other molecules. But knowing the factorization of a number gives you limited information about how it “reacts” with other numbers. For instance, knowing the factorizations of x and y tells me absolutely nothing about the factorization of x + y.
You understand correctly, by the way - C[sub]2[/sub]H[sub]5[/sub]OH is nothing like CH[sub]3[/sub]OCH[sub]3[/sub] despite having the same empirical formula C[sub]2[/sub]H[sub]6[/sub]O, to give a simple but fairly drastic example.
I’m just wondering what the isotopes of 7 are…
There are some relationships between physics and prime numbers. I recently read “Stalking the Riemann Hypothesis: The Quest to Find the Hidden Law of Prime Numbers” by Dan Rockmore. I liked it a lot. Near the end it covers some of the eerier coincidences (or maybe not) between primes and certain properties of atoms.
I think you’re missing a few ellipses there.
What a clusterfuck !
3
If you wanted to stretch the analogy, you could say either variations in the unit or in the representation of the quantity. (7 apples vs. 7 orangutans; 7/1 versus 2+5)
But that sorta raises a deeper question: where do the subatomic particles show up in number theory? Aren’t quantities… uh… ‘indivisible’? Maybe prime numbers as protons and neutrons makes more sense.
And where would you find zero and infinity in a chemistry text?
To me primes are holes in the arithmetic, not building blocks. You can’t get 17 by multiplying anything. It’s a hole in the multiplication table.
Probably says more about me than about math, though.
Tris
Okay, here’s the proof. The proposition is the following: the n-ary expansion of a rational number x terminates if and only if x = p/q where gcd(p,q) = 1 and the prime factors of q are all factors of n.
Suppose that the n-ary expansion of x terminates. Then we can write
x = m + sum(a_k n^(-k), k=1…l) with m an integer, l in {0, 1, 2, …} and a_k in {0, 1, …, n-1}
llll= (m n^l)/n^l + sum((a_k n^(l-k))/n^l, k=1…l)
llll= (m n^l + sum(a_k n^(l-k), k=1…l))/n^l.
We can rewrite the preceding fraction as an irreducible fraction x = p/q, and all prime factors of q will be factors of n^l and therefore of n.
Now suppose that we can write x = p/q where gcd(p,q) = 1 and where all prime factors of q are factors of n. Then there exists an integer l such that n^l x = m an integer, which we rewrite as x = m/n^l = m n^(-l). This means that the n-ary expansion of x ends at the l-th position after the point (or before).
What am I missing here (or did I get whooshed)? Last I heard, 66=36 so that number ends in 6, not 9 and 147=98 so that number has 98 repeating.
.14141414… = 14/99, not 1/7.
The OP asks if prime numbers are like “elements”, not like the chemical elements that compose the periodic table. He only mentions the chemical elements as an aside, after the question.
So, literally, yes, and not just “like”. In fact prime numbers ARE elements, they are elemental and irreduceable building blocks, of integer products.
We’ve discussed the matter of repeating decimals many times. Here’s a thread which contains links to other threads and to a report on the subject:
If you search on the words “repeating” and “decimals” in the SDBM archives you’ll find other threads about the subject.
I appreciate that you’re exploring, but I can’t see any serious relationship. 
A broken car cannot turn into a parking lot.
Eggs can turn into an omelette but an omelette cannot turn into eggs.
Bricks are basic building blocks of houses.
LEGO is the basic building block of a theme park.
In chemistry, since by definition it only deals with matter, zero could the point where matter becomes energy. Infinity might be a black hole.
What you’re missing in the first case is that the number doesn’t end at all, so you just go on seeing the carries (3 + 6 = 9) forever, never hitting a terminal 6. In the second case, it wasn’t your fault, but, as ultrafilter points out, .14141414… isn’t actually 1/7.
>In chemistry, since by definition it only deals with matter, zero could the point where matter becomes energy. Infinity might be a black hole.
“Physics” is generally defined as the study of matter, and is probably the scientific field that most fundamentally deals with energy and certainly black holes. Chemistry is more the study of interactions between chemical elements and chemical compounds, isn’t it?