Mathlover
Reputation
5,694
Top tag
Next privilege 10,000 Rep.
Access moderator tools
 Apr 19 comment Understanding Leibniz formula: $D^n (f g) = \sum\limits_{k=0}^{n} \binom{n}{k} D^{n-k}f D^kg$ @LukasArvidsson You are welcome. I am very glad that my answer helped you. Feb 29 comment Finding Divisibility of Sequence of Numbers Generated Recursively @MarkusScheuer Great work. As you pointed out, there is big similarity with my question to prove the divisibility , Your method also worked in this question well to obtain $x^k$ terms because of $e^x$ terms. Really ,Your series expansion method is very powerful to analyze divisibility for similar problems . Thanks a lot for sharing . Best Regards Feb 26 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer I have noticed the wrong typing now even if I am very careful about it. I apologize for wrong typing your last name in my previous comment. I cannot edit it because İt is disabled. If you want, I can delete it. Thanks a lot for your understanding Feb 26 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer: I have not found any book or link that shows that relation $n^{d}\equiv P(n,d) \pmod {d}$. I just wondered if you found any link about it. Please share if you found a link. Best Regards Feb 25 comment Solving differential equation $y''(x)+Q(x)y(x)=0$ If you know any particular solution $y_p(x)$ that solves $y''(x)+Q(x)y(x)=0$ , You will find the general solution of your equation. There is no closed known solution method to find the particular solution if $Q(x)$ is not given. Of course there are a lot of endless series methods exist to solve the equation but no closed formula for general $Q(x)$. Feb 25 comment Solving differential equation $y''(x)+Q(x)y(x)=0$ If you do not know an particular solution $(y_1(x))$ for the equation , there is no known general solution for the equation for $Q(x)$. If you want to check my question that related to this problem. math.stackexchange.com/questions/99850/… Feb 25 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner Your last bonus opens a huge mysterical door for new research field. I have been searching the internet for $$n^d \equiv P(n,d) \pmod {d}$$ but I have not found the theory. Have you found a related work in somewhere ? Best Regards Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer Thank you for your advice and help . I added an answer that shows the proof of my conjecture. Please advice if there is a mistake in it. Best Regards Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuer : Thank you a lot for your answer. It gave me a great point how to find the general formula of $P(n,d)$. Now I am testing it for some values. Best Regards Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner : And then easily $A_n(x)=(e^x - x)^n$ can be gotten. It is very nice result Feb 23 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner : Wonderful relations . Please check the relation while applying the recurrence relation in (4) . I think it should be $$\sum_{d=0}^\infty (P(n+1,d)+d.P(n,d-1))\frac{x^d}{d!}\\$$. then we can get $$A_{n+1}(x)=(e^x-x)A_{n}(x)$$ Feb 21 comment How many different numbers can be written if each used digit symbol is used at least 2 times? @MarkusScheuner : I believe that $P(n,d)$ function can be a key function to detect prime numbers. It has wonderful features Feb 21 comment How many different numbers can be written if each used digit symbol is used at least 2 times? Your answer is great. Have you read my conjecture : $n^{d}\equiv P(n,d) \pmod {d}$ , I have not found any counter-example yet. It satisfies for any number in your table but I do not know how to prove it .Thanks for helps. Feb 21 comment How many different numbers can be written if each used digit symbol is used at least 2 times? :Thanks a lot for answer . I edited my question with wonderful discover for $P(n,d)$. Could you please share your comments on my last edit? Feb 21 comment How many different numbers can be written if each used digit symbol is used at least 2 times? Thanks a lot for answer . I edited my question with wonderful discover for $P(n,d)$. Could you please share your comments on my last edit? Jan 15 comment What is lower limit condition of a surface of a tetrahedron? Can we have only one lower condition ? or impossible. If it exists, it should be cyclic .for example, the lower condition could be \$|S_2-S_3|+|S_3-S_4|+|S_2-S_4|