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2h
revised Exponent of an exponent?
adding the tag [exponentiation]
2h
suggested approved edit on Exponent of an exponent?
2h
answered Exponent of an exponent?
1d
answered Is it true a matrix $A$ has determinant $0$ if and only if $A^N=0$?
1d
comment Decide what is the number of roots of the equation
And by plugging $x=0$, $x=5$ and $x=10$ (and the limits for $x\to -\infty$, $x\to +\infty$) into $f$ and into $g(x):=100x$ (which is very easy to do without electronic aid) we see that the solution which is close to $0$, is positive, and the solution which is close to $10$, is a less than $10$.
Aug
28
comment How many distinct roots $ax^5+bx^3+cx+d$ has
Another way to say this is that $f$ has an inverse $f^{-1}:\mathbb{R}\to\mathbb{R}$, so $f(x)=0$ has the unique solution $x=f^{-1}(0)$.
Aug
28
comment Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$
A somewhat small example for $k=6$ with $n=5$ "stretches" of nines is, explicitly, $$009999999999999999999999999999999998\\ 999999999999999999999999999999999998\\ 999999999999999999999999999999999998\\ 999999999999999999999999999999999998\\ 999999999999999999999999999999999999$$ This is just under $10^{5 \cdot 36 - 2} = 10^{178}$. Your formula gave two figures zero at the end which I moved to the front.
Aug
28
comment Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$
I could find many examples for $k=6$ with your first formula. Such as $(m,n,i)=(10,9,29)$ which gives $10^{261}-10\sum_{s=0}^8 10^{29s}$. This number has a digit sum of $2332$ while its sixth power has a digit sum of only $2305$. It seems you are right this produces infinitely many examples for $k=6$, so this is really interesting. However it is hard to say what the smallest for $k=6$ is (I mean examples that do not come from your formula)? Is it your opinion that one can find explicit examples for $k=5$ with the program your propose?
Aug
24
comment A circle in the plane contains at most four lattice points?
@MarcvanLeeuwen Google found a book (by W. Sierpinski, some pages available for free from Google Books) where they give explicit circles with exactly 0, 1 and 2 rational points. Then they complete "my" proof that a circle with more rational points than that must have an infinity of them. They translate to center $(0,0)$. Then since there exists one rational solution $(a,b)$ to $x^2+y^2=r^2$, they use that $a$ and $b$ to write $x=\frac{2at+b(1-t^2)}{1+t^2}$; $y=\frac{a(1-t^2)-2bt}{1+t^2}$, so just like in M. Bennet's answer below. They remark that the rational points on it form a dense subset.
Aug
24
comment A circle in the plane contains at most four lattice points?
@MarcvanLeeuwen Interesting. By lattice points I suppose you mean points in $\mathbb{Z}\times\mathbb{Z}$? Do you know what natural numbers $n$ are attainable as lattice points (in this sense) on a circle (whose center can have irrational coordinate(s))? If instead we count the number of points in $\mathbb{Q}\times\mathbb{Q}$, my argument shows (I think) that if that number is finite but exceeds $2$, then it is a multiple of $4$. But do you know what numbers $n$ are attainable as the number of $\mathbb{Q}\times\mathbb{Q}$ points on a circle?
Aug
20
comment The smallest integer whose digit sum is larger than that of its cube?
Sorry, I had a typo. It is sumdigits. I do not know what version of PARI/GP it requires. But it makes your code a lot faster. Only supports base 10, though.
Aug
19
comment The smallest integer whose digit sum is larger than that of its cube?
(your last comment) Yes! (That is a typo for A064209.)
Aug
18
comment Find all local maximum and minimum of $f$, which is $1$ if the decimal expansion contains a $5$ and $0$ otherwise.
@CarlHeckman But the question mentions a situation where the expansion ends in "an infinite string of 9s".
Aug
18
comment Find all local maximum and minimum of $f$, which is $1$ if the decimal expansion contains a $5$ and $0$ otherwise.
If $f$ well-defined? If $x$ has two representations, for example $x=0.75000\ldots = 0.74999\ldots$, then what is $f(x)$? One or zero?
Aug
18
awarded  Nice Question
Aug
18
comment Squares of a number yields a palindrome?
Also see (Wonderful) Demlo numbers.
Aug
18
comment The smallest integer whose digit sum is larger than that of its cube?
@MartinR I just found out (from A261439) that sumdigts(n) is a built in function in PARI/GP. Makes the search faster.
Aug
18
comment For which positive integers n does there exist a prime whose digits sum to n?
Another question that comes to my mind is: Is there any integer $n$ which is the digit sum of infinitely many distinct primes? For example if $n=2$ were to have this property, it would imply an infinitude of generalized Fermat primes of the form $10^{2^m}+1$ which goes against usual conjectures on such primes.
Aug
18
comment The smallest integer whose digit sum is larger than that of its cube?
@PatrickStevens and others, I asked this as a separate question (should appear in "Linked" threads at the right).
Aug
18
asked Integers $n$ for which the digit sum of $n$ exceeds the digit sum of $n^5$