When is ${b^2 - {b-1}_2}_2=1$ with odd $b$? (The bracket-notation explained below)

The article of L. Szalay as mentioned by user Random, gives the solution.
We can reformulate $$ begin{array} {} {b^2 - {b-1}_2 }_2 & overset{?}= 1 &&& (1.1)\ b^2 - {b-1}_2 & overset{?}= 2^A & &&(1.2)\ 2^B b^2 - (b-1) & overset{?}= 2^B 2^A & &&(1.3)\ 4 cdot 2^{2B} b^2 - 4cdot 2^Bb + 1 & overset{?}= 2^{2B+2} 2^A &- 2^{B+2}&+1& (1.4)\ (2cdot 2^Bb - 1)^2 & overset{?}= 2^{A+2B+2} &- 2^{B+2}&+1& (1.5)\ hline x^2 & = 2^n &- 2^m &+1& (2)\ end{array}$$ Szalay gives the following solutions in theorem 2:

• $ displaystyle (n,m, x)in { (2t,t+1,2^t-1) mid t in mathbb N, t ge 2 }$.
  Inserting this gives by $x=2^t-1$ that $b=1$, which is of course a trivial solution for $f(b)=1$.

• $ displaystyle (n,m, x) in {(t,t,1) mid t in mathbb N, t ge 1 }$
  Here, to have $x=1$ we must have $2cdot 2^Bb - 1=1$ and $ 2^Bb =1 implies b=1, B=0$ which is again the trivial solution, and

• $ displaystyle (n,m, x) in {(5,3,5),(7,3,11),(15,3,181) }$
  For $x=5$ we get $b=3$ which is the "known" solution, for $x=11$ we get again $b=3$ and finally for $x=181$ we get $b=91$ . By the latter we have in the lhs that $B=0$ but in the rhs we have with $m=3$ that $B=1$, which is in contradiction to the specific form in which my equation realizes Szalay's (with additional restrictions on $x$), so this is not an allowed case.

Conclusion: the two cases $b=1$ (trivial) and $b=3$ (known) are the only solutions for the equation in question.

Underlying problem solved: For the generalized Collatz-problem $ T(x): x to {mx+1 }_2 $ this means:

there are exactly the following $2$-step-cycles with $b=T(a)$ and $a=T(b)$ when $a=1$ is fixed:

• for $m$ of the form $m=2^k-1$ the trivial 2-step-cycle $a=b=1$ and
• for $m=5$ and $(a,b,m)=(1,3,5)$

(The known 2-step-cycles for $m=181$ are not counted because their smallest element $a$ is $a ne 1$)

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^{Thanks to user "Random" for the reference to Szalay's article.}

For Q1:

Let $b=2^kt+1$ with odd $t$. Then equality $f(b)=1$ is equivalent to $$(2^kt+1)^2-t=2^m$$ for some $m$ (notice that $m>2k$). This is a quadratic equation in $t$. To have integer solutions, its discriminant must be a square, that is $$2^{2k+2+m}-2^{k+2}+ 1=z^2$$ for some $z>0$.

There are no solutions with even $m$ since $$(2^{k+1+m/2})^2 > 2^{2k+2+m}-2^{k+2}+ 1 > (2^{k+1+m/2}-1)^2.$$ For odd $m$, we have $$2^{k+2} - 1 = 2^{2k+2+m}-z^2 = (2^{k+1+(m-1)/2}sqrt{2}-z)(2^{k+1+(m-1)/2}sqrt{2}+z) > (2^{k+1+(m-1)/2}sqrt{2}-z)2^{k+2+(m-1)/2}(sqrt{2}+1),$$ implying that $$0 < sqrt{2}-frac{z}{2^{k+1+(m-1)/2}} < frac{1}{2^{k+m-1}(sqrt{2}+1)} < frac{1}{(2^{k+1+(m-1)/2})^{3/2}}.$$ That is, $frac{z}{2^{k+1+(m-1)/2}}$ represents an order $3/2$ rational approximation to $sqrt{2}$, which suggests (although this is just a heuristics) that we should look for such fractions among convergents and semiconvergents to $sqrt{2}$. Their denominators are given by Pell numbers (A000129) and sums of consecutive Pell numbers (A001333). The latter are odd, while it is easy to establish that the former contain just two powers of $2$ -- namely, $1$ and $2$.

PS. The known solution gives $z=11$, $k=1$, $m=3$ with the rational approximation $frac{11}{8}$ to $sqrt{2}$ not being a convergent or a semiconvergent. So, above is not a proof but a heuristic argument that such approximations are extremely rare, and cannot be obtained from the known sources (convergents and semiconvergents) for good rational approximations.

For Q2, if $p equiv 1 bmod 3$, then $p$ may be written as $p=x^2+xy+y^2$ for some $x,yinmathbb{Z}$. Additionally if $b^2-b+1=p^m$ for some integer $m>0$, then it may be seen that $b notequiv 1 bmod p$ and hence ${b-1}_p=b-1$. The above gives $(x^2+xy+y^2)^m=b^2-b+1$ or $$ ((x-yomega)(x-yomega^2))^m=(b+omega)(b+omega^2) $$ where $omega=exp(2 pi i/3)$. For example, in your example, $$ 7=2^2+2+1 $$ and $$ 7^3=19^2-19+1 $$ as $(2-omega)^3$ times a unit of $mathbb{Z}[omega]$ is equal to $19+omega$.

Also, for Q1, a partial answer (that can probably be extended further) is:

Let $1leq a_1 < a_2 < dots < a_n$ and let

$$ b=1+sum_{i=1}^n 2^{a_i}. $$

If $a_{n-1} < a_n - 2$, then $b-2^{a_n} < 2^{a_{n-1}+1} leq 2^{a_n-2}$ and $b^2 - 2^{2a_n} < 2^{2a_n}$. Thus if we write $b^2$ as

$$ b^2 = 2^{2a_n} + sum_{i in A}2^i, $$

the largest $i$ in $A$ satisfies $a_n+a_{n-1}+1 leq i < 2a_n$. However, for such $b$, we would have to have

$$ b^2 = 2^{2a_n} + sum_{i=1}^n 2^{a_n-a_1} $$

if ${b^2 - {b-1}_2}_2 = 1$. However, $a_n-a_1 < a_n+a_{n-1}+1$. Hence $b$ with $a_{n-1} < a_n - 2$ cannot satisfy ${b^2 - {b-1}_2}_2 = 1$.

Write

$$ b = 1+sum_{i=1}^n 2^{a_i} = 2^{a_n} + a $$ and assume that $a>0$. Then

$$ b^2 = (2^{a_n} + a)^2 geq 2^{2a_n+1} $$

if and only if

$$ a^2 + 2^{a_n+1}a - 2^{2a_n} geq 0. $$

As $a geq 0$, that means that

$$ b^2 geq 2^{2a_n+1} Leftrightarrow a geq 2^{a_n}(sqrt{2}-1). $$

If ${b^2 - {b-1}_2}_2 =1$, then $$ b^2 = 2^q + sum_{i=1}^{n}2^{a_i-a_1} $$

where $$ q=begin{cases}2a_n+1 mbox{ if } a geq 2^{a_n}(sqrt{2}-1)\2a_nmbox{ otherwise.}end{cases} $$

It is the case that

$$ {b-1}_2 = sum_{i=1}^n 2^{a_i-a_1} < 2^{a_n}. $$

If $a < 2^{a_n}(sqrt{2}-1)$ then $$b^2 - 2^{2a_n} = 2^{a_n+1}a + a^2 > 2^{a_n}.$$ Therefore if $a < 2^{a_n}(sqrt{2}-1)$, we cannot have ${b^2 - {b-1}_2}_2 =1$. Also, if $a geq 2^{a_n-1}$, then

$$ b^2 - 2^{2a_n+1} geq 2^{2a_n-2}. $$

$2^{2a_n-2} > 2^{a_n}$ if $a_n > 2$. Therefore if either $a < 2^{a_n}(sqrt{2}-1)$ or ($a geq 2^{a_n-1}$ and $a_n > 2$), we cannot have ${b^2 - {b-1}_2}_2 =1$.

Using this approach, it is possible to obtain:

Writing $b = 2^{a_n} + a$, with $0 < a < 2^{a_n}$, then ${b^2 - {b-1}_2}_2 = 1$ cannot happen when $a in mathbb{Z}$ and

$$ frac{a}{2^{a_n}} in (0,sqrt{2}-1) cup [sqrt{2+frac{1}{2^{a_n}}}-1,1). $$

The length of the interval $[2^{a_n}(sqrt{2}-1),2^{a_n}(sqrt{2+frac{1}{2^{a_n}}}-1))$ converges to $frac{1}{2sqrt{2}}$ as $a_n rightarrow infty$ and hence can contain at most one integer for each $a_n geq 1$. Hence for each $a_n geq 1$, there is at most one integer $a$ with $0 < a < 2^{a_n}$ such that $b = 2^{a_n} + a$ could possibly satisfy ${b^2 - {b-1}_2}_2 = 1$.