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I noticed that there are a lot of equations in physics with factors of 2, 3 and 5 (either in the numerator or in the denominator), but there aren't any with factors of 7 or any prime number greater than 7.

It's like the first three prime numbers (2, 3 and 5) are "more important" than the other prime numbers in physics. What is the reason for this?

Gurn64
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Quantum Force
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    Lots of physics equations involve integration or differentiation; when you have 3 dimensions, this means that factors of 2 and 3 can show up readily. A 5 shows up in the moment of inertia of a sphere (you integrate over 3 dimensions with an object that changes shape in 2 dimensions). It is possible to come up with shapes that would have a moment of inertia with a 7 in the equation - it's just not a "simple" shape. – Floris May 29 '14 at 18:26
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    No number is "more important" than 17, which is proven to be the world's most random number. But, yeah, physics tries to start simple --"assume a spherical cow" -- so larger constants and larger primes don't show up until you get into longer series expansions or more dimensions. – Carl Witthoft May 29 '14 at 18:56
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    possible duplicate of this question – Flint72 May 29 '14 at 20:35
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    @CarlWitthoft: Clearly 4 is the most random number – Kyle Kanos May 29 '14 at 20:36
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    Concerning the prime 11, see this famous formula for the beta function. – Qmechanic May 29 '14 at 21:01
  • This question appears to be off-topic because it is about arbitrary factors for arbitrary equations with arbitrary units. – Brandon Enright May 29 '14 at 21:26
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    @KyleKanos Prior to the event you cite, it was well-established that 17 is the most random number. I guess it's hard for us old guys to accept new results sometimes. – garyp May 29 '14 at 22:47
  • @BrandonEnright yet right in your comment, there's clear use of the prime "3" . – Carl Witthoft May 30 '14 at 11:58

1 Answers1

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There are several reasons why squares and cubes occur more often than powers of 11, 7, 17, or 4. They are: reduction, apparent three dimensional space of our world, and "simple" relations in our models (laws) of the universe.

Reduction

We tend to reduce equations into their simplest form. We prefer to cancel out powers and reduce fractions whenever possible, to make it easier to calculate and remember. Even though there are infinite powers divisible by any number, 2's and 3's divide into other numbers more easily than other numbers. Go ahead a pick a number. It will likely be more often divisible by 2 or 3 rather than 17 or 11 or 7. Even if you choose random numerators and denominators in a fraction, and attempted to reduce them, you will likely see more factors of 2 or 3 than these other primes.

Apparent Three Dimensional Space

We appear to live with three dimensions of space. Obviously, any laws which depend on position need to have three variables, and laws the depend on an area need to have two. Our math system tends to work well with cartesian coordinates, which allows us to compare dimensions on equal footing, allowing us to square and cube our dimensions. I suspect (but have not attempted) to survey all known physical laws in their spherical forms, which may reduce the apparent number of 2's and 3's.

Three dimensional space also gives rise to inverse-square laws, such as those which govern the force of gravity or force from an electric field on an object. If we lived in a universe with more or less spacial dimensions, we'd see different powers in these equations.

Simple Relations

Most of the models we use to describe the universe have found simple relations between things. This was simple practicality on the part of physicists, but also a lucky break. Sure, you can force students and thought-experiments to proceed by giving them basic information (like a table of position-time pairs), but you could give them slightly more advance information (like velocity) and simplify their lives as well as yours.

Beyond this explanation... I can only say it just so happens to be this way.

PipperChip
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    With due respect, the last sentence is a more believable answer as compared to the previous three points. :) – 299792458 May 30 '14 at 09:12