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December 31

Close powers of odd numbers.

Having just seen that 13^3=2197 and 3^7=2187, I was wondering if there are known examples as close or closer at higher numbers. More specifically, p,q,m & n, such that p and q are odd primes, m &n are 2 or larger. p^m>2200 and p^m-q^n<=10.Naraht (talk) 04:50, 31 December 2020 (UTC)[reply]

No examples are known and there are none with p^m < 10¹⁸. I don't know whether it has been proved there are no solutions. See Catalan's conjecture#Generalization and http://www.sspectra.com/Pillai.txt. PrimeHunter (talk) 09:17, 31 December 2020 (UTC)[reply]

It is still an open problem. See also the generalized Tijdeman problem. In contrast to the absolute difference, you can get the ratio as close to 1 as you want. In this case, 13³/3⁷ = 1.00457... . But 13²⁴²/3⁵⁶⁵ = 1.00180... . --Lambiam 10:12, 31 December 2020 (UTC)[reply]

A relevant (very technical) article: Jan Turk (1986). "On the difference between perfect powers" (PDF). Acta Arithmetica. 45: 289–307. A much easier read: Differences Between Powers. --Lambiam 10:36, 31 December 2020 (UTC)[reply]

I don't know what is known beyond 10¹⁸ but I made a script which searched p^m < 10³⁰ without finding solutions. PrimeHunter (talk) 13:40, 31 December 2020 (UTC)[reply]

Work and Wages

A, B and C completed work costing $ 1800. A worked for 6 days, B for 4 days and C for 9 days. If their daily wages are in the ratio of 5:6:4. How much amount will be received by A?

Solution: Daily wages = 5 : 6 : 4 => 6*5 : 5*6 : 9*4 = 30:24:36

A : B : C = 5 : 4 : 6

A's amount = 5/15*1800 = 600$

Why we have to calculate A : B : C again if daily wage already mentioned in question? Rizosome (talk) 06:33, 31 December 2020 (UTC)[reply]

@Rizosome:

Please do your own homework.

Welcome to Wikipedia. Your question appears to be a homework question. I apologize if this is a misinterpretation, but it is our aim here not to do people's homework for them, but to merely aid them in doing it themselves. Letting someone else do your homework does not help you learn nearly as much as doing it yourself. Please attempt to solve the problem or answer the question yourself first. If you need help with a specific part of your homework, feel free to tell us where you are stuck and ask for help. If you need help grasping the concept of a problem, by all means let us know.--Jasper Deng (talk) 07:44, 31 December 2020 (UTC)[reply]

@Jasper Deng: In what way does it look like homework to you? Rizosome (talk) 11:49, 31 December 2020 (UTC)[reply]

The OP has the solution, but does not understand the reasoning. --Lambiam 08:54, 31 December 2020 (UTC)[reply]

To start, in the line with "Solution:", you wrote "5*6". That should have been "4*6" (4 days times 6). The number "24" that follows is correct.

There are two different ratios involved. The first one, 5 : 6 : 4, is for the daily wages of each of the three workers. It does not depend on how long each worked. The second, 30 : 24 : 36 = 5 : 4 : 6, is for the amount they will receive for the work that each performed. This one does depend on how long each worked. The $ 1800 must be divided among the workers in this ratio, and not in the other ratio. The number 600$ for A is the same either way, but that is by accident. For B and C it does make a difference. --Lambiam 08:54, 31 December 2020 (UTC)[reply]

How can I know "m" in final step refer to 600$?

2 Men and 1 women together can complete a piece of work in 14 days, while 4 women and 2 men together can do it in 8 days. If a man gets $ 600 per day, how much should a woman get per day?

Solution: 2m + 1 w = 1/14 -> 28m + 14w = 1 (eq: 1)

2m + 4w = 1/8 -> 16m + 32w = 1 (eq: 2)

28m + 14w = 16m + 32w

12m = 18w

1w = 2/3m = 2/3 (600) = 400$.

How can I know "m" in final step refer to 600$? Rizosome (talk) 11:47, 31 December 2020 (UTC)[reply]

Damn, that's a sexist exercise. It apparently assumes all men work equally fast, and all women work equally fast but slower and should be paid accordingly. Are you sure it isn't a trick question where the wanted answer is "The same as a man"? Anyway, before the last line, m and w refer to which fraction of the work a man and woman can do in one day. The last part "= 2/3 (600) = 400$" is either lazy writing or suddenly changes the meaning of w and m to what they should get per day.PrimeHunter (talk) 12:10, 31 December 2020 (UTC)[reply]

The men all have families to support, and you have to assume that the women will all quit when they get married :) --RDBury (talk) 12:38, 31 December 2020 (UTC)[reply]

(edit conflict) The letters

m

and

w

in the initial equations stand for the amount of work done per day (as a fraction of completing a piece of work). This can be expressed as a ratio

m : w = ? : ?

. There seems to be an unstated assumption that the daily wage should be proportional to the amount of work per day. In the last line just before the question, the letter

w

in "

1w = ...

" suddenly stands for something else, namely the daily wage of a woman. This is confusing; it is better to use a different letter. If

x

stands for the daily wage of a woman, we get

m : w = ? : ? = 600$ : x

. Here

m

does not refer to the wage. --Lambiam 12:40, 31 December 2020 (UTC)[reply]