# If $q$ is prime, can $\sigma(q^{k-1})$ and $\sigma(q^k)/2$ be both squares when $q \equiv 1 \pmod 4$ and $k \equiv 1 \pmod 4$?

This is related to this earlier MSE question. In particular, it appears that there is already a proof for the equivalence $$\sigma(q^{k-1}) \text{ is a square } \iff k = 1.$$

Let $$\sigma(x)$$ denote the sum of divisors of the positive integer $$x$$.

Here is my question:

If $$q$$ is prime, can $$\sigma(q^{k-1})$$ and $$\sigma(q^k)/2$$ be both squares when $$q \equiv 1 \pmod 4$$ and $$k \equiv 1 \pmod 4$$?

MY ATTEMPT

Suppose that $$\sigma(q^{k-1}) = a^2$$ and $$\frac{\sigma(q^k)}{2} = b^2$$ for $$q \equiv 1 \pmod 4$$ and $$k \equiv 1 \pmod 4$$.

Since $$\sigma(q^k) = q^k + \sigma(q^{k-1})$$, it follows that $$2b^2 = \sigma(q^k) = q^k + \sigma(q^{k-1}) = q^k + a^2.$$

Additionally, congruence-wise we obtain $$a^2 = \sigma(q^{k-1}) \equiv 1 + (k-1) \equiv k \equiv 1 \pmod 4,$$ from which it follows that $$a$$ is odd, and $$2b^2 = \sigma(q^k) = q^k + \sigma(q^{k-1}) \equiv 1^1 + 1 \equiv 2 \pmod 4,$$ which implies that $$b$$ is likewise odd.

Now, using the definition of $$\sigma(q^k)$$ and $$\sigma(q^{k-1})$$ for $$q$$ prime, we derive $$\frac{1}{2}\cdot\frac{q^{k+1} - 1}{q - 1} = b^2$$ and $$\frac{q^k - 1}{q - 1} = a^2.$$

Assume to the contrary that $$\frac{1}{2}\cdot\frac{q^{k+1} - 1}{q - 1} = b^2 \leq a^2 = \frac{q^k - 1}{q - 1}.$$ This assumption leads to $$q^{k+1} - 1 \leq 2(q^k - 1)$$ which implies that $$16 = {5^1}(5-2) + 1 \leq q^k(q - 2) + 1 = q^{k+1} - 2q^k + 1 \leq 0,$$ since $$q$$ is a prime satisfying $$q \equiv k \equiv 1 \pmod 4$$. This results in the contradiction $$16 \leq 0$$. Consequently, we conclude that $$a < b$$.

Furthermore, I know that $$(q+1) = \sigma(q) \mid \sigma(q^k) = 2b^2$$ so that $$\frac{q+1}{2} \leq b^2.$$

Finally, I also have $$\frac{q^{k+1} - 1}{2b^2} = q - 1 = \frac{q^k - 1}{a^2}.$$

Alas, here is where I get stuck.

CONJECTURE (Open)

If $$q$$ is a prime satisfying $$q \equiv k \equiv 1 \pmod 4$$, then $$\sigma(q^{k-1})$$ and $$\sigma(q^k)/2$$ are both squares when $$k = 1$$.

SUMMARY OF RESULTS SO FAR

zongxiangyi appears to have proven the implication $$\sigma(q^k)/2 \text{ is a square} \implies k = 1.$$

The proof of the following implication is trivial $$k = 1 \implies \sigma(q^{k-1}) \text{ is a square}.$$ The truth value of the following implication is currently unknown: $$\sigma(q^{k-1}) \text{ is a square} \implies k = 1.$$

Together, the two results give $$\sigma(q^k)/2 \text{ is a square} \implies k = 1 \iff \sigma(q^{k-1}) \text{ is a square},$$ so that $$\sigma(q^{k-1})$$ is a square if $$\sigma(q^k)/2$$ is a square.

Therefore, $$\sigma(q^{k-1})$$ and $$\sigma(q^k)/2$$ are both squares (given $$q \equiv 1 \pmod 4$$ and $$k \equiv 1 \pmod 4$$) when $$\sigma(q^k)/2$$ is a square.

• I guess, If the conjecture holds, then $q\equiv 1 \pmod{2^n}$ for $n\ge 1$ such that $q>2^n$ . – Zongxiang Yi Jul 6 at 7:19
• From the equation $$\sigma(q^k)/2 = b^2$$ I can derive $$\bigg(\frac{q^{(k+1)/2} + 1}{2}\bigg)\cdot\bigg(\frac{q^{(k+1)/2} - 1}{q - 1}\bigg) = b^2.$$ Note that $$\gcd\bigg(\frac{q^{(k+1)/2} + 1}{2},\frac{q^{(k+1)/2} - 1}{q - 1}\bigg)=1.$$ – Jose Arnaldo Bebita-Dris Jul 7 at 3:35
• Therefore, if $\sigma(q^k)/2$ is a square, then both $$\frac{q^{(k+1)/2} + 1}{2}$$ and $$\frac{q^{(k+1)/2} - 1}{q - 1}$$ are squares. In particular, $$\sigma(q^{(k-1)/2}) = \frac{q^{(k+1)/2} - 1}{q - 1}$$ is a square. So we now have the (somewhat stringent) requirement that $$\sigma(q^k)/2, \sigma(q^{k-1}), \text{ and } \sigma(q^{(k-1)/2})$$ are all squares. – Jose Arnaldo Bebita-Dris Jul 7 at 3:45

Here are a couple of other approaches to consider which may be useful. First, your equation of

$$2b^2 = \sigma(q^k) = q^k + \sigma(q^{k-1}) = q^k + a^2 \tag{1}\label{eq1}$$

can be rewritten as

$$2b^2 - a^2 = q^k \tag{2}\label{eq2}$$

This is in the generalized Pell equation form of $$x^2 - Dy^2 = N$$. The blog Solving the generalized Pell equation explains how to solve this.

Next, note that

$$\sigma(q^{k-1}) = \sum_{i=0}^{k-1} q^i \tag{3}\label{eq3}$$

$$\sigma(q^{k}) = \sum_{i=0}^{k} q^i \tag{4}\label{eq4}$$

Thus, you can express $$\sigma(q^{k})$$ in terms of $$\sigma(q^{k-1})$$ as

$$\sigma(q^{k}) = q\sigma(q^{k-1}) + 1 \tag{5}\label{eq5}$$

As you stated, suppose

$$\sigma(q^{k-1}) = a^2 \tag{6}\label{eq6}$$

$$\frac{\sigma(q^k)}{2} = b^2 \iff \sigma(q^k) = 2b^2 \tag{7}\label{eq7}$$

Substituting \eqref{eq6} and \eqref{eq7} into \eqref{eq5} gives

$$2b^2 = qa^2 + 1 \iff 2b^2 - qa^2 = 1 \iff (2b)^2 - (2q)a^2 = 2 \tag{8}\label{eq8}$$

Wikipedia's Pell's equation page's Transformations section gives a related equation of

$$u^{2}-dv^{2}=\pm 2 \tag{9}\label{eq9}$$

and how it can be transformed into the Pell equation form of

$$(u^{2}\mp 1)^{2}-d(uv)^{2}=1 \tag{10}\label{eq10}$$

Here, $$u = 2b$$, $$v = a$$, $$d = 2q$$ and the right side of \eqref{eq8} is $$2$$, so \eqref{eq10} becomes

$$((2b)^2 - 1)^2 - (2q)(2ba)^2 = 1 \tag{11}\label{eq11}$$

This is in Pell's equation form of $$x^2 - ny^2 = 1$$. Since $$n = 2q$$ is not a perfect square, there are infinitely many integer solutions. However, among these solutions, you first need to check that $$x$$ is in the form $$4b^2 - 1$$, the determined $$b$$ divides $$y = 2ba$$ and then that $$a$$ and $$b$$ satisfy \eqref{eq6} and \eqref{eq7} for some $$k \equiv 1 \pmod 4$$.

As for your open conjecture, if $$k = 1$$, then isn't $$\sigma(q^{k-1}) = \sigma(q^{0}) = 1$$ and $$\frac{\sigma(q^{k})}{2} = \frac{\sigma(q)}{2} = \frac{1 + q}{2}$$, so having both of them be squares requires $$q = 2b^2 - 1$$ for some $$b$$ and, thus, is not always true for all primes $$q \equiv 1 \mod 4$$, e.g., for $$q = 5$$, you get $$5 = 2b^2 - 1 \implies 6 = 2b^2 \implies b = \sqrt{3}$$?

• Thank you for your answer. Regarding your assertion in the last paragraph of your answer: Does the restriction $q = 2b^2 - 1$ for some $b$ lead to a contradiction? I doubt that it does, @JohnOmielan. – Jose Arnaldo Bebita-Dris Jul 6 at 13:38
• @JoseArnaldoBebita-Dris I've given a very simple example where it doesn't work for all primes $q \equiv 1 \pmod 4$, i.e., $q = 5$. Am I misunderstanding something? – John Omielan Jul 6 at 16:21
• Okay, I get your point @JohnOmielan. See zongxiangyi's proof for the implication $$\sigma(q^k)/2 \text{ is a square} \implies k = 1$$ and my own proof for the equivalence $$\sigma(q^{k-1}) \text{ is a square} \iff k = 1.$$ – Jose Arnaldo Bebita-Dris Jul 6 at 16:24
• I hereby retract my claim in the previous comment. Work is currently underway to prove the implication $$\sigma(q^{k-1}) \text{ is a square} \implies k = 1.$$ (The proof of the converse is trivial.) – Jose Arnaldo Bebita-Dris Jul 6 at 21:07

(This proof is currently under reconstruction.)

Let $$q$$ be a prime satisfying $$q \equiv k \equiv 1 \pmod 4$$.

I (attempt to) prove here that

$$\sigma(q^{k-1}) = s(q^k) \text{ is a square } \implies k = 1.$$

Proof

Assume to the contrary that $$k > 1$$. This implies that $$k \geq 5$$ (since $$k \equiv 1 \pmod 4$$).

Suppose that $$s(q^k) = s^2 = \sigma(q^k) - q^k = \sigma(q^{k-1}) = \frac{q^k - 1}{q - 1}.\tag{*}$$

$$(*)$$ implies that $$(q-1)s^2 = q^k - 1$$, which is equivalent to $$q(q^{k-1} - s^2) = q^k - qs^2 = 1 - s^2 = (1 + s)(1 - s) = -(s+1)(s-1).$$

Since $$q$$ is prime, we consider three two cases:

Case 1: $$q \mid s + 1$$

SubCase 1.1: $$q = s + 1$$ $$\implies q - 1 = s \implies q^3 - 3q^2 + 3q - 1 = (q - 1)^3 = s^3 = (q - 1)s^2 = q^k - 1$$ $$\implies q^2 - 3q + 3 = q^{k-1} \geq q^4$$ This last inequality is a contradiction.

SubCase 1.2: $$q < s + 1$$

Take $$1 < r = (s+1)/q$$. Then from the equation $$q(s^2 - q^{k-1}) = (s+1)(s-1)$$ one gets $$s^2 - q^{k-1} = r(s - 1)$$ so that $$(s - 1) \mid (s^2 - q^{k-1}) = \sigma(q^{k-2})$$ where $$s - 1 = \sqrt{\sigma(q^{k-1})} - 1$$.
This implies that $$(s - 1) \nmid q^{k-1}$$ since $$(s - 1) \mid (s^2 - q^{k-1})$$ and $$\gcd(s-1,s)=1$$. In particular, $$(s - 1) \nmid q^{k-1}$$ implies that $$s \notin \left\{2, q+1, \ldots, q^{k-1} + 1\right\},$$ since the only possible divisors of $$q^{k-1}$$ are $$1, q, \ldots, q^{k-1}$$. But $$q \mid (s+1)$$. (No contradictions thus far.)

Note that $$\sigma(q^{k-2}) \equiv 1 + (k - 2) \equiv k - 1 \equiv 0 \pmod 4.$$ Also, we have the inequality $$\sqrt{\sigma(q^{k-1})} - 1 = s - 1 < s^2 - q^{k-1} = \sigma(q^{k-2}).$$ This last inequality implies that $$\sqrt{\frac{q^k - 1}{q - 1}} < \frac{q^{k-1} - 1}{q - 1} + 1 = \frac{q^{k-1} + q - 2}{q - 1}$$ from which we get $$\frac{q^k - 1}{q - 1} < \bigg(\frac{q^{k-1} + q - 2}{q - 1}\bigg)^2$$ which means that $$(q^k - 1)(q - 1) < (q^{k-1} + q - 2)^2.$$ (I am currently unable to get a contradiction under this SubCase 1.2.)

Case 2: $$q \mid s - 1$$

SubCase 2.1: $$q = s - 1$$ $$\implies q + 1 = s \implies q^{k-1} = s^2 - s - 1 = (q+1)^2 - (q+1) - 1$$ $$= q^2 + 2q + 1 - q - 1 - 1 = q^2 + q - 1$$ $$\implies q^2 + q - 1 = q^{k-1} \geq q^4$$ Again, this last inequality is a contradiction.

SubCase 2.2: $$q < s - 1$$

Take $$1 < t = (s-1)/q$$. Then from the equation $$q(s^2 - q^{k-1}) = (s+1)(s-1)$$ one gets $$s^2 - q^{k-1} = t(s + 1)$$ so that $$(s + 1) \mid (s^2 - q^{k-1}) = \sigma(q^{k-2})$$ where $$s + 1 = \sqrt{\sigma(q^{k-1})} + 1$$.

This implies that $$(s + 1) \nmid q^{k-1}$$ since $$(s + 1) \mid (s^2 - q^{k-1})$$ and $$\gcd(s,s+1)=1$$. In particular, $$(s + 1) \nmid q^{k-1}$$ implies that $$s \notin \left\{q-1, \ldots, q^{k-1} - 1\right\},$$ since the only possible divisors of $$q^{k-1}$$ are $$1, q, \ldots, q^{k-1}$$. But $$q \mid (s-1)$$. (No contradictions thus far.)

Note that $$\sigma(q^{k-2}) \equiv 1 + (k - 2) \equiv k - 1 \equiv 0 \pmod 4.$$ Also, we have the inequality $$\sqrt{\sigma(q^{k-1})} + 1 = s + 1 < s^2 - q^{k-1} = \sigma(q^{k-2}).$$ This last inequality implies that $$\sqrt{\frac{q^k - 1}{q - 1}} < \frac{q^{k-1} - 1}{q - 1} - 1 = \frac{q^{k-1} - q }{q - 1}$$ from which we get $$\frac{q^k - 1}{q - 1} < \bigg(\frac{q^{k-1} - q}{q - 1}\bigg)^2$$ which means that $$(q^k - 1)(q - 1) < (q^{k-1} - q)^2.$$ (I am currently unable to get a contradiction under this SubCase 2.2.)

QED

In fact, more is true.

If $$k=1$$, then $$s(q^k) \text{ is a square}$$.

Therefore, we have the biconditional $$s(q^k) = \sigma(q^{k-1}) = \frac{q^k - 1}{q - 1}$$ is a square if and only if $$k=1$$.

• The proof can be simplified by observing that $s(q^k) = \sigma(q^{k-1}) = s^2 \equiv 1 + (k-1) \equiv k \equiv 1 \pmod 4$ implies that $s$ is odd. The three cases (i) $q = s + 1$, (ii) $q = s - 1$, and (iii) $q = (s+1)(s-1)$ then all contradict $q \equiv 1 \pmod 4$. – Jose Arnaldo Bebita-Dris Jul 6 at 15:46
• Please see the updated answer, which is currently under (re)construction. – Jose Arnaldo Bebita-Dris Jul 6 at 22:39