Number with same digit sum of only ones in multiple bases Let $n$, $k$, $a_1$ and $a_2$ be positive integers.
Does there exist $k$ such that


*

*There are exactly $n$ ones in it.

*There are any number of zeroes in it.

*There exists no digit but zeroes and ones.


whether written in number system of base $a_1$ or base $a_2$?
Note that $n$ has constant value, so both representations must have same number of ones. Also, the conditions will always be met for base $2$, so most likely $a_1=2$.
If no, is there a simple proof? If yes, is there a generalized method for finding such numbers, and have they been studied before?
 A: Let $a,b,n$ be positive integers such that $a<b$. We use the notation $S(a,b,n)$ for the set of integers which have exactly $n$ number of ones in both the base $a$ and in the base $b$ and no other digits. as a first observation we have :
$$S(a,a^m,n)=\left \{\sum_{i∈A}a^{im} \Big /A\subset \mathbb{N},\#(A)=n\right \}$$
(The sum is taken over all elements in $\#(A)$ denotes the cardinal of $A$)
Now if we return to the definition, finding an element $k\in S(a,b,n)$ is equivalent to solving an exponential Diophantine equation of the form:
$$a^{i_1}+⋯+a^{i_n}=b^{j_1}+⋯+b^{j_n}\\ i_1<i_2<\cdots<i_n\\
j_1<j_2<\cdots<j_n$$
As an example,for $n=2, a=2,b=3$ the equation to solve is 
$$2^x+2^y=3^u+3^v$$
this equation have a solution which is $(2,3,1,2)$? Equations of this form are not always easy to solve and in particular when we deal with the same bases I mean there is $n$ powers of $a$ and $n$ powers of $b$, the very known useful too is eﬃcient congruencing method which can very helpful to solve a lot of equations, But if we return to your question we know that (using only congruence methods):

Theorem $(1)$ : Let p, q be distinct odd primes such that $q = 1 \mod p$. Then the equation
   $$p^a + p^b = q^c + q^d$$
  have the only trivial solution.

which signifies $S(p,q,2)=\emptyset $ in this case, this implies that there are some cases for $n$ where there is no $k$ which verifies your question. we can also prove easily that $S(a,ab,n)=\emptyset $ using some elementary divisibility problems , And in general one can study this set and deduce some properties using only elementary methods.
$(1)$ PACIFIC JOURNAL OF MATHEMATICS Vol. 101, No. 2, 1982
EXPONENTIAL DIOPHANTINE EQUATIONS
J. L. BRENNER AND LORRAINE L. FOSTER
