Wikipedia:Reference desk/Archives/Mathematics/2016 June 21

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June 21

Equation of plane

Can i find equation of a plane by having known the angles it makes with three axis.if required i know one point on that plane also? SD — Preceding unsigned comment added by Sameerdubey.sbp (talk • contribs) 05:54, 21 June 2016 (UTC)

Knowing the angles gives you a family of parallel planes, so you do need one other datum such as a point on the plane. Off the top of my head, this might work: ±sin(α)x ± sin(β)y ± sin(γ)z = k, where the three angles are (as I assume you intend) with the plane rather than with its normal vector, and k is determined by the known point. The plus and minus depend on which side of the plane is "up"; I hope someone else can express that more clearly. If the angles are with the normal vector, replace sin with cos. —Tamfang (talk) 05:58, 26 June 2016 (UTC)

Arithmetic sequences classified by whether they are closed under multiplication

Do you know of any formula to use in determining whether an arithmetic sequence whose terms are integers is closed under multiplication. Such sequences can be classified by its first term (a) and the difference (d.) Note that a must be in the interval from 0 to d-1. Let me see how I can do for low values of d:

• d = 1: Yes
• d = 2: Yes, yes
• d = 3: Yes, yes, no

So we have our first no; this is d=3 and a=2, which means the sequence 2, 5, 8, 11... The definition of the no is that it is not closed under multiplication; 2 times 5 is 10. Now, let me see if there's any pattern when it comes to extending to higher values of d:

• d = 4: Yes, yes, no, no

If I'm correct, all those where a=0 or 1 are closed under multiplication. At this point we might conjecture that all those where a is in the 2 to d-1 interval are not. Does d = 5 disprove it yet??

• d = 5: Yes, yes, no, no, no
• d = 6: Yes, yes, no, yes, yes, no

So it appears that d=5 doesn't disprove this conjecture, but d=6 does! The sequences 3,9,15,21,27... and 4,10,16,22,28,34,40... are closed under multiplication. Any set rule that can be used in determining whether an arithmetic sequence is closed under multiplication without actually calculating whether the product of 2 terms is in the sequence?? Georgia guy (talk) 14:32, 21 June 2016 (UTC)

Can you clarify what you mean by "closed under multiplication"? Because at first I thought you mean that if S=s_n is a sequence, then S_k=k*s_n is a subsequence of S, for any k. That would match up with your d=2 being "yes, yes" on closure, and d=3, a=2 case being "no". But then 3,9,15,21,27... and 4,10,16,22,28,34,40... are not closed, for the same reason, but you say they are? So I'm not sure if there's some typos, or I'm being dense, etc., but something is not... adding up for me :) SemanticMantis (talk) 14:46, 21 June 2016 (UTC)

It means that for all a and b in a sequence, then a times b is also in the sequence. Georgia guy (talk) 14:49, 21 June 2016 (UTC)

Thanks. So $S=s_{n}$ is "closed under multiplication" iff for any j, for any k, $s_{j}\cdot s_{k}\in S$, or alternatively, for any sequences $J_{n},K_{n}\in \mathbb {N}$, then $s_{J_{n}}\cdot s_{K_{n}}$ is a subsequence of S. I don't know off hand, how to check this, but at least know we know what you mean. SemanticMantis (talk) 14:55, 21 June 2016 (UTC)

We have $x_{n}=a+nd$ and $x_{k}=a+kd.$ Then $x_{n}x_{k}$ is in the sequence iff $x_{n}x_{k}-a$ is divisible by d, which it is iff $a(a-1)$ is divisible by d. It is when a equals 0 or 1. I'll think about whether any other cases work. Loraof (talk) 17:57, 21 June 2016 (UTC)

The case $d=a(a-1)$ obviously works. E.g., a=3, d=6; or a=7, d=42. Any others? Loraof (talk) 18:07, 21 June 2016 (UTC)

I believe it works iff a(a–1) = qd for integer q ≥0, since qd is divisible by d. Loraof (talk) 18:13, 21 June 2016 (UTC)

In other words we're looking for pairs (d, a): d|a(a-1). There seems to be a number of parametric solutions, for example (n(n-1), n), (n(n-1)/2, n), (2n-1, 4n-2), (2n, 4n-2), (n(n+1), n²) but I don't see a pattern for a general solution. There is a symmetry in that (d, a) is a solution iff (d, d+1-a) is a solution. --RDBury (talk) 21:19, 21 June 2016 (UTC)

Although a couple of those violate the condition that d > a. Loraof (talk) 22:50, 21 June 2016 (UTC)

Oops, I meant to write (4n-2, 2n-1), (4n-2, 2n) for two of them. A two parameter solution is (m(mn-1), mn) and it's symmetrical partner is (m(mn-1), (m-1)(mn-1)). This includes four of those single parameter solutions but not (n(n-1)/2, n). --RDBury (talk) 05:40, 22 June 2016 (UTC)

Equivalently, a has to be an idempotent under multiplication mod d. The number of solutions mod d is therefore $2^{\omega (d)}$, where $\omega (d)$ is the number of distinct prime factors of d. GeoffreyT2000 (talk) 16:23, 22 June 2016 (UTC)

Mathematical quotation

A famous mathematician, referring to a notoriously complicated function, once said something along the lines of "If a powerful alien species threatened to destroy the Earth unless we calculated F(a), we should dedicate all the world's computing power to it. If they ordered us to calculate F(b), we should fight them." What is the function? The name of the mathematician and the values of a and b will presumably be trivial to determine, given this information. Tevildo (talk) 21:08, 21 June 2016 (UTC)

Erdős said it about Ramsey numbers. --RDBury (talk) 21:23, 21 June 2016 (UTC)

That's it, thanks. Tevildo (talk) 21:43, 21 June 2016 (UTC)