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May 24

About the prime values of the sigma function and Ramanujan tau function

Let $\sigma (n)$ be the sum-of-divisors function (sequence A000203 in the OEIS) and $\tau (n)$ be the Ramanujan tau function (sequence A000594 in the OEIS).

1. If $\sigma (n)$ or/and $\tau (n)$ is prime, must $n$ be of the form $p^{q-1}$ with $p,q$ both primes?

2. Does there exist a positive integer $n$ such that $\sigma (n)$ and $\tau (n)$ are both primes? If so, do there exist infinitely many such positive integers $n$? 218.187.70.249 (talk) 04:05, 24 May 2023 (UTC)

The answer to the first half of the first question (i.e. for $\sigma (n)$) is yes. See OEIS A023194, particularly the last comment. GalacticShoe (talk) 04:24, 24 May 2023 (UTC)

The answer to the second half of the first question (i.e. for $\tau (n)$) is probably yes. In particular, if $\tau (n)$ is an odd prime, then $n$ is indeed of the specified form. Whether $|\tau (n)|=2$ has any solutions is an open problem. See OEIS A135430. GalacticShoe (talk) 04:34, 24 May 2023 (UTC)

The answer to the first half of the second question is yes. The first few terms of the sequence of $n$ for which $\sigma (n)$ and $\tau (n)$ are both primes are $458329$, $46117681$, $49182169$, $56957209$, $104509729$, $\ldots$. GalacticShoe (talk) 04:42, 24 May 2023 (UTC)

Currently how many such positive integers $n$ (i.e. $\sigma (n)$ and $\tau (n)$ are both primes) are known? (Could you create an OEIS sequence for such $n$?) Are there any known large (> 10¹⁰⁰⁰) such $n$? (For large $n$, unproven probable primes $\sigma (n)$ and $\tau (n)$ are allowed, also negative primes $\tau (n)$ are allowed) 218.187.70.249 (talk) 05:08, 24 May 2023 (UTC)

First question is hard to quantify, since there doesn't seem to have been any literature studying this particular sequence of integers. As for the second question, Lygeros and Rozier's paper mentioned in A135430 has a list of large $n$ for which $\tau (n)$ is prime or probably prime; I was not able to personally confirm whether any of those $n$ have the property that $\sigma (n)$ is prime or probably prime due to their sheer sizes. GalacticShoe (talk) 13:54, 28 May 2023 (UTC)

It took less than a minute to test the values in their table 3 with OpenPFGW. The only cases where $\sigma (p^{q-1})$ is also prime are 677² = 458329, and 947⁴⁰. $\tau (n)$ is negative in both cases but the absolute value is prime. PrimeHunter (talk) 15:19, 28 May 2023 (UTC)

I have computed nine solutions above 10¹⁰⁰⁰ with PARI/GP and OpenPFGW. PARI/GP verification:.

v=[1236727^190, 3283991^180, 3666881^180, 11534591^172, 8051963^180,\
   9224857^180, 12432997^178, 18623489^180, 7723841^192];
for(i=1, #v, n=v[i]; s=sigma(n); t=abs(ramanujantau(n));\
print([#digits(n), #digits(s), #digits(t), ispseudoprime(s), ispseudoprime(t)]))

Output:

[1158, 1158, 6366, 1, 1]
[1173, 1173, 6451, 1, 1]
[1182, 1182, 6500, 1, 1]
[1215, 1215, 6681, 1, 1]
[1244, 1244, 6837, 1, 1]
[1254, 1254, 6896, 1, 1]
[1263, 1263, 6946, 1, 1]
[1309, 1309, 7198, 1, 1]
[1323, 1323, 7274, 1, 1]

ispseudoprime makes a strong probable prime test. $\tau (n)$ is positive for 1236727¹⁹⁰, 3666881¹⁸⁰, 12432997¹⁷⁸, and negative for the other six. PrimeHunter (talk) 02:02, 29 May 2023 (UTC)

OK, additional questions:

1. If $p$ is prime, is there always a prime $q$ such that $\sigma (p^{q-1})$ is prime? If so, are there infinitely many such primes $q$?
2. If $p$ is prime, is there always a prime $q$ such that $\tau (p^{q-1})$ is prime? If so, are there infinitely many such primes $q$?
3. If $q$ is prime, is there always a prime $p$ such that $\sigma (p^{q-1})$ is prime? If so, are there infinitely many such primes $p$?
4. If $q$ is prime, is there always a prime $p$ such that $\tau (p^{q-1})$ is prime? If so, are there infinitely many such primes $p$?
5. For each prime $p<1000$, find the smallest prime $q$ such that $\sigma (p^{q-1})$ is prime.
6. For each prime $p<1000$, find the smallest prime $q$ such that $\tau (p^{q-1})$ is prime.
7. For each prime $q<1000$, find the smallest prime $p$ such that $\sigma (p^{q-1})$ is prime.
8. For each prime $q<1000$, find the smallest prime $p$ such that $\tau (p^{q-1})$ is prime.

211.75.79.246 (talk) 06:38, 29 May 2023 (UTC)

Note: In questions 5 to 8, we allow strong probable primes > 10¹⁰⁰⁰. 211.75.79.246 (talk) 06:45, 29 May 2023 (UTC)

Questions 1 through 4 imply that there are infinitely many numbers for which $\sigma (n)$ or $\tau (n)$ are prime, and are thus harder than the question of whether there are infinitely many numbers for which $\sigma (n)$ or $\tau (n)$ are prime. As with many problems of the form "are there infinitely many numbers such that [some property regarding primality]", I expect that the answer is "probably yes, but no one has proven it yet." The best that can be done is heuristic predictions. Lygeros and Rozier's paper mentioned earlier, for example, gives an estimation on the probability that $\tau (p^{q-1})$ is prime. GalacticShoe (talk) 14:22, 29 May 2023 (UTC)

For question 5, see OEIS A065854. GalacticShoe (talk) 14:29, 29 May 2023 (UTC)

For question 7, see OEIS A123487. GalacticShoe (talk) 14:39, 29 May 2023 (UTC)

Areas in projection of regular icosahedron

Assume an regular icosahedron of edge 1 unit with each side colored differently, resting on the X-Y plane. A plane at Z=10 (or any height completely above the icosahedron) is colored based on the color of the Icosahedron on the face directly below it. This will have 10 colors. The Central triangle has area root(3)/4 , the others are a set of three triangles which share an edge with the central triangle and then six even more distorted that only share a vertex. What are the areas of the other triangles? I'm thinking the three that share an edge would be the area of the central times the absolute value of the cosine of the dihedral angle between sides (which according to the article is 138.189685° = arccos(−^√5⁄₃)), but I have no idea how to figure out the area of the projection of the other six triangles.Naraht (talk) 09:17, 24 May 2023 (UTC)

If you have the $(x,y,z)$ coordinates of the vertices of any of the triangles, the vertices of its orthogonal projection on any z-plane are obtained by leaving out the $z$ coordinate, resulting in a set of three $(x,y)$ coordinates. Several formulas for the area of a triangle can then be used.  --Lambiam 16:15, 24 May 2023 (UTC)

At Regular icosahedron § Cartesian coordinates you can find the coordinates of the vertices (for edge length = 2). By ignoring the z-coords, these will give you the vertices of the projected triangles. Pick a triangle of each type, one of whose edges is parallel to an axis (eg identical x-coords on two vertices), and 1/2 base x height will give the area. To check your math, the total projected area should equal the area of a hexagon with the same circumradius as the icosahedron. -- Verbarson  ^talk_edits 16:22, 24 May 2023 (UTC)

A potential alternative approach is based on the fact that the area of the orthogonal projection of any planar shape – not only polygons – is equal to the original area times the absolute value of the cosine of the angle between the two planes involved. Once you know the angles the polyhedral faces make with the z plane, you're basically done.  --Lambiam 16:42, 24 May 2023 (UTC)

This was mentioned by the OP, but only for three of the triangles. The article Projected area mostly covers the cosine rule, though the article seems to be a stub and needs a lot of work. So basically this amounts to finding the cosines of the angles at the origin between the vertices of the dual, aka the dodecahedron. This can be done easily using dot products. --RDBury (talk) 16:56, 24 May 2023 (UTC)

The cosine for the "not quite adjacent" triangles is 1/3. I did this by direct calculation but in hindsight it should have been clear because of the relationship between the dodecahedron and the cube (or the relationship between the icosohedron and the octahedron). You can pick 8 vertices from a dodecahedron which form a cube, or dually, you can pick 8 faces from an icosohedron which, when extended, form an octahedron. So really the angle between the faces is the same as the angle between adjacent faces in the octahedron. The relevant coordinates involve only integers so the calculations are much easier. --RDBury (talk) 17:51, 24 May 2023 (UTC)

May 25

What's the difference between a mathematical formula and a mathematical equation?

On StackOverflow, an user says that:

An equation is any expression with an equals sign, so your example is by definition an equation. Equations appear frequently in mathematics because mathematicians love to use equal signs.

A formula is a set of instructions for creating a desired result. Non-mathematical examples include such things as chemical formulas (two H and one O make H2O), or the formula for Coca-Cola (which is just a list of ingredients). You can argue that these examples are not equations, in the sense that hydrogen and oxygen are not "equal" to water, yet you can use them to make water.

If that's the case, then is there mathematical formulas that are not equations, or mathematical equations that are not formulas? If you make an equation for something, then by definition you would want the equation to give a desired result/do a desired action, so I'm not sure of the difference between the two terms here. CactiStaccingCrane (talk) 17:27, 25 May 2023 (UTC)

The difference is pragmatic. Purely formally, an equation consists of two expressions connected by an equals sign. One typically uses the term formula for an equation that gives the relation between an entity whose value is desired to be known, and other entities whose values it depends on, expressed as an equation. An example is the formula for the area $T$ of a triangle in terms of its height $h$ and the length $b$ of its base:

$T={\frac {1}{2}}bh.$

Another example is the formula for the solution of the linear equation $ax+b=0,$ where $a\neq 0$:

$x=-{\frac {b}{a}}.$

One can think of such formulas as representing conclusions of theorems. Equations also appear in building a mathematical model for a problem, such as, "In a cylindrical barrel with radius 2 m, the water is rising at a rate of 3 cm/s. What is the rate of increase of the volume of water?" (Note that the assumptions that the barrel is standing upright and is not leaky are not stated.) Letting $L$ stand for the water level in cm and $t$ for the elapsed time in seconds, we have $L=3t,$ and so on. One would not call these equations "formulas". Likewise for the quadratic equation $x^{2}+x=1$. These equations are not like the conclusion of some theorem.  --Lambiam 20:22, 25 May 2023 (UTC)

I should add that in informal use any slightly complex mathematical expression may be referred to as a "mathematical formula".^[1][2][3]  --Lambiam 07:07, 26 May 2023 (UTC)

I guess there are going to be a lot of different takes on this; both terms are part of the language used to describe mathematics and neither really has a formal definition. (There is a definition of 'equality' and the meaning of '=' itself in set theory, but that's not the same thing as an equation.) I would say an equation is a mathematical statement or condition that involves equality. An equation can be either true (e.g. 2+2=4) or false (e.g. 2+2=5). Frequently whether an equation is true depends on the value of some variable. For example 2x+10=4x+6 is an equation which is true or false depending on the value of x. Solving an equation is the process of finding values of the variable for which the equation is true, for example solving the previous equation produces x=2. A formula is an expression used to produce a desired result. For example the quadratic formula

${\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

can be used to solve a quadratic equation ax²+bx+c=0. You can solve an equation which has an unknown variable, but there's no such thing as solving a formula. A formula will often be given as an equation, as in

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

with the desired result on the left and the expression used to find that result on the right, but it's not strictly necessary. (Though there are probably a few math teachers who would deduct a point or two for not including "x=" in the quadratic formula on an exam answer.) But the difference is rather fuzzy since you can certainly treat a formula given as an equation like any other equation. For example A=πr² is the formula for the area of a circle in terms of it's radius. But you can solve for r and produce an equivalent formula $r={\sqrt {\frac {A}{\pi }}}$ for the radius terms of the area. Like I said at the start, these terms are part of the language of mathematics with no formal definition, so whether you consider something an equation, a formula, or neither can depend on context or what you're actually doing with it.

It might be worth mentioning that a statement involving <, >, ≤ or ≥ is often called an inequality. Like an equation, an inequality involving an unknown variable can be solved, though the process is different. --RDBury (talk) 07:39, 26 May 2023 (UTC)

May 27

How many unit hyperballs can touch a unit hypersphere simultaneously without overlap(s)?

Is it pi**2*2 round down? And the surface hypervolume of the unit hyperhypersphere round down for five dimensions and so on? Sagittarian Milky Way (talk) 16:06, 27 May 2023 (UTC)

The number of unit spheres that can simultaneously touch another unit sphere is the Kissing number. The article has numbers (sometimes just a range) for dimensions up to 24. AndrewWTaylor (talk) 17:12, 27 May 2023 (UTC)

So the 2, 2pi (~6.28), 4pi (~12.57), 2pi^2 (~19.74) thing completely collapses after enough degrees of freedom (I didn't know that number shrinks forever after ~33.07 for dimension 7) Sagittarian Milky Way (talk) 18:53, 27 May 2023 (UTC)

May 29

Prove or disprove

Prove or disprove: All even numbers >=4 can be written as $2^{m}+3^{n}+p$ with m>=0, n>=0, and p prime. 61.224.134.44 (talk) 11:20, 29 May 2023 (UTC)

OEIS:A303702 gives the number of solutions for each even number and conjectures there are always solutions. It states a low search limit of 3×10⁹. PrimeHunter (talk) 12:59, 29 May 2023 (UTC)

Prove or disprove:

1. All odd numbers >3 can be written as sum of “twice a triangular number” (>0) and a prime.
2. All odd numbers >3 can be written as sum of “twice a square number” (not necessary >0, the square number 0 is allowed, or 17 cannot be written as this form) and a prime.
3. All odd numbers >3 can be written as sum of “twice a generalized pentagonal number” (>0) and a prime.

2001:B042:4005:47FA:78CF:2178:A4F1:85F0 (talk) 13:39, 29 May 2023 (UTC)

For question 1, see OEIS A144590. No numbers are known that cannot be written as such. GalacticShoe (talk) 15:03, 29 May 2023 (UTC)

For question 2, see OEIS A046923. In particular, note that 5777 and 5993 cannot be written as two times a square plus a prime. Testing up to 1,000,000 reveals no other numbers that cannot be written as such. GalacticShoe (talk) 14:56, 29 May 2023 (UTC)

The theory behind AI

What is the mathematical theory behind AI? Graph theory or statistics? 2A02:908:424:9D60:2463:DEAC:9F20:EBA9 (talk) 16:55, 29 May 2023 (UTC)

Um. Big subject. Have you tried reading Artificial intelligence for starters to get a basic grounding? NadVolum (talk) 17:07, 29 May 2023 (UTC)

If one must select a single area of mathematics (not theory), mathematical optimization is IMO the best (most relevant) choice. There is no unified theory underlying AI.  --Lambiam 17:38, 29 May 2023 (UTC)

This video series, by 3blue1brown, is an excellent overview of the mathematics of artificial neural networks, which form one of the most common ways that machine learning and artificial intelligence works. It provides a surface-level overview of the mathematics, but in a way that gives you plenty of opportunities for further study. --Jayron32 17:51, 30 May 2023 (UTC)

May 30

Deriving the square root of seven from a unit polycube

Distances between vertices of a double unit cube are square roots of the first six natural numbers, including the reference desk/mathematics (√7 is not possible due to Legendre's three-square theorem)

Based on File:distances_between_double_cube_corners.png, I drew an illustration of ways to get square roots of 1 to 6 from a polycube. Adding a third cube to make an L shape gives √8 and √9=3, whereas adding it in-line gives √9, √10 and √11.

Is there any way to get √7?

Thanks,
cmɢʟee⎆τaʟκ 00:10, 30 May 2023 (UTC)

The way that these lengths are obtained is by taking the square root of the sums of squares of lengths of the "bounding box" of the line segment. For example, ${\sqrt {6}}={\sqrt {2^{2}+1^{2}+1^{2}}}$ derives from two sides of length $1$ and a side of length $2$. Since there are three sides to the bounding box, the only lengths obtainable are precisely those that can be written as the square root of the sums of three squares. $7$, quite famously, is not one of those numbers. GalacticShoe (talk) 01:39, 30 May 2023 (UTC)

Note that this implies that all ${\sqrt {n}}$ for nonnegative $n$ can be obtained if working with four-dimensional polycubes. GalacticShoe (talk) 01:46, 30 May 2023 (UTC)

Thank you very much, @GalacticShoe. Glad to learn about Legendre's three-square theorem and Lagrange's four-square theorem. Cheers, cmɢʟee⎆τaʟκ 08:27, 30 May 2023 (UTC)