About Some Relatives of Palindromes

Andrei T\"or\"ok; Viorel Nitica

REVIEW 3 major objections 5 minor 9 references

About Some Relatives of Palindromes

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper defines two new classes of integers and proves that all palindromes with an even number of digits or an even middle digit are in the first class, while all squares of palindromes with at least two digits are in the second.

desk verdict New digit-sum classes with a clean additive half, but the headline square-of-palindromes theorem is not proved as printed; the two-digit case has a false inequality at a=b−1. read the letter →

arxiv 1908.00713 v1 pith:ZOIJPWGZ submitted 2019-08-02 math.NT

Viorel Nitica , Andrei Török This is my paper · ORCID

classification math.NT MSC 11B8311B99

keywords b-wARHnumbersMRHpalindromedigitsumreversalnumerationbaseb-NivenRamanujan-Hardy

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces two families of integers, b-wARH and b-wMRH numbers, defined by asking that a number $N$ can be reproduced from the sum of its own base-b digits plus an extra term $A$, either by adding that sum's reversal (wARH) or by multiplying by it (wMRH). Its main results are inclusion theorems: every palindrome with an even number of digits, and every palindrome with an odd number of digits and an even middle digit, is b-wARH, and every square of a palindrome with at least two digits is b-wMRH, for every base $b\ge 2$. These families strictly contain the previously studied b-ARH and b-MRH numbers. The paper also proves there are infinitely many b-wARH and b-wMRH numbers in every base, gives bounds on how many digits such numbers can have for a fixed extra term, and relates the new classes to b-Niven numbers.

What carries the argument

The load-bearing object is the extra term $A$, a nonnegative integer satisfying $N = A+s_b(N)+(A+s_b(N))^R$ for b-wARH numbers and $N=(A+s_b(N))((A+s_b(N))^R)$ for b-wMRH numbers. The proofs work by choosing $A$ so that $A+s_b(N)$ becomes a prescribed number whose reversal is already known: the zero-padded half of the palindrome for the additive class, and the palindrome itself for the multiplicative class. The analytic ingredient that keeps $A$ nonnegative is Proposition 5, especially the digit-sum inequality $s_b(P^2)\le P$ for integers with at least three base-b digits.

What would settle it

For the two-digit palindromes the paper leaves to numerical checking, $P=[22]_6$, $P=[22]_7$, and $P=[22]_8$, find all integers $X$ with $X\cdot X^R=P^2$ and check whether any of them satisfies $X\ge s_b(P^2)$. If none does, Proposition 17(a) fails for that base; a single such case would refute the claim that all squares of palindromes with at least two digits are b-wMRH numbers.

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Extended reading notes

Core claim

The central discovery is that a single extra-term construction turns digit sums into reversals that rebuild a number. For an even-length palindrome $N$ whose digits are $a_1\ldots a_n\,a_n\ldots a_1$, the choice $A=[a_1\ldots a_n\,0^n]_b - s_b(N)$ makes $A+s_b(N)$ equal to the zero-padded left half of $N$, whose reversal is the right half; adding them reproduces $N$. For a palindrome $P$ with at least two digits, the choice $A=P-s_b(P^2)$ makes $A+s_b(P^2)=P$, so $P^2$ is the product of $P$ with its own reversal. The same equations give the inclusions of the old classes: any b-ARH number is b-wARH and any b-MRH number is b-wMRH, using $A=s_b(N)(M-1)$. The paper claims these two inclusion statements hold in every base $b\ge 2$.

Load-bearing premise

The proof that every square of a palindrome with at least two digits is a b-wMRH number rests on the digit-sum inequality $s_b(P^2)\le P$; if this fails for some palindrome, the chosen extra term $A=P-s_b(P^2)$ would be negative and the construction would not give a witness.

Editorial extensions

If this is right

Every base-b palindrome of even length, and every base-b palindrome of odd length with an even middle digit, is a b-wARH number for every base $b\ge 2$.
Every base-b square of a palindrome with at least two digits is a b-wMRH number; more generally, if $P$ has at least three base-b digits, the same construction shows $P\cdot P^R$ is b-wMRH.
Every b-ARH number is b-wARH and every b-MRH number is b-wMRH, with extra term $A=s_b(N)(M-1)$, so the new classes strictly contain the old ones.
For every base $b\ge 2$ there are infinitely many b-wARH numbers, infinitely many b-wMRH numbers, and infinitely many corresponding extra terms.
There are infinitely many b-wARH numbers that are not b-MRH numbers and infinitely many b-wMRH numbers that are not b-MRH numbers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction behind Proposition 17(a) actually works for any product $P\cdot P^R$ when $P$ has at least three base-b digits, not only for palindromic squares; the paper's statement singles out palindromes, but the same choice of $A$ proves the wider claim for this product-with-reversal family.
Because the digit-sum inequality is the only analytic input, the inclusion results may be sensitive to base only through the size of $s_b$; bases with slack in the inequality are likely to admit many alternative extra terms, which connects to the paper's open multiplicity questions.
A systematic scan of $s_b(P^2)-P$ over all two-digit palindromes would settle the finite cases the paper leaves to numerical checking and could reveal whether a uniform proof exists for all bases.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

What you should know: this paper introduces two new integer classes, b-wARH and b-wMRH, and claims all suitable palindromes are in the first and all squares of palindromes with at least two digits are in the second. The definitions are natural variants of the authors' earlier b-ARH and b-MRH classes, and the additive half is clean. The multiplicative half, however, is not proved as written.

The additive result, Proposition 7(a), is genuinely nice. The construction chooses A as a shifted palindrome minus the digit sum, and Proposition 5(a) ensures nonnegativity. That part works. The paper also has useful corollaries about divisors and non-Niven examples, and the digit-length bounds in Theorems 38–43 look plausible.

The trouble is Proposition 17(a), the square-of-palindromes result. For two-digit P=[aa]_b with a^2 ≥ b, the proof bounds s_b(P^2) by 4a^2 − 4b + 4 and then claims inequality (13) holds at the endpoints. At a=b−1 the left side is 3(b−1)(b−3), not the printed 6−7b, and it is positive for every b>3. So the proof fails for infinitely many two-digit palindromes, not just the finite cases left as numeric checks. The underlying claim may still be true — [33]_4 works, for example — but the printed argument does not establish it. This is a load-bearing gap because the abstract's second headline depends on exactly this proposition.

There are also mechanical errors that look like typos but as printed are false. Proposition 16 gives A_k = b^{2k}−2 for N_k = [1 0^{2k} 1]_b, but the defining equation needs A_k = b^{2k+1}−2. Proposition 23 gives A_k = b^k−1, which is off by two; the calculation needs b^k−3. The table counts for the two classes are swapped (77 vs 365), and some small two-digit subcases are dismissed as "can be checked numerically."

The additive half and the general framework are solid enough that this paper deserves a serious referee, but the referee needs to redo the two-digit computation. If the case-3 gap is fixed with a sharper carry estimate, the paper is publishable as an elementary number theory note. As it stands, I would not cite it yet. My recommendation: send it to peer review, but flag the specific hole in Proposition 17(a) so the referee actually checks the arithmetic.

Referee Report

3 major / 5 minor

Summary. The paper introduces two new classes of integers, b-wARH and b-wMRH numbers, which relax the earlier b-ARH and b-MRH definitions by replacing the multiplier with an extra term A. The main claims are that every palindrome with an even number of digits, or with an odd number of digits and even middle digit, is b-wARH, and that every square of a palindrome with at least two digits is b-wMRH. The proofs rest on digit-sum inequalities in Proposition 5, a case analysis for two-digit palindromes, and several corollaries about infinitude, divisibility, and digit-length bounds. The paper also contains computational tables and OEIS references.

Significance. If established, the two inclusion theorems are attractive and give a large, easily described supply of numbers in the new classes. The lemma s_b(P^2) <= P for palindromic P is a useful observation, and Corollary 20, stating that every integer divides some b-wMRH number, is striking. The computational tables and OEIS entries are valuable additions. However, the manuscript as written does not prove the central inclusions: the main two-digit case for Proposition 17(a) contains a false inequality, Proposition 5(a) is false as stated, and the proof of Proposition 5(c) has a digit-count gap. These issues are load-bearing for the abstract's claims, so the paper needs substantial revision before the results can be accepted.

major comments (3)

[§6, Case 3, Eq. (13)] The proof of Case 3 in Proposition 17(a) is invalid. The quadratic in (13), f(a)=4a^2-a(b+1)+4(1-b), evaluates at a=b-1 to 3(b-1)(b-3), not the printed "6-7b"; this is positive for every b>3. Hence the claimed inequality fails at a=b-1, so the upper bound s_b(P^2) <= 4a^2-4b+4 is not shown to be at most P. Consequently the argument does not establish s_b(P^2) <= P for the infinitely many two-digit palindromes [b-1,b-1]_b. The underlying statement may still be true, but the proof as written does not prove it.
[§3, Prop. 5(c), Eqs. (11)-(12)] The proof of Proposition 5(c) asserts that N^2 has 2n-1 digits, but for N=b^n-1 one has N^2=b^{2n}-2b^n+1, which has 2n digits; for example, 999^2=998001 in base 10. The digit-sum bound should therefore be (b-1)*2n, and the printed inequality (12), with 2n-1, does not suffice. The text also states that (12) holds for n=3 and b>=2, but for b=2,n=3 it reads 5<=4, forcing the subsequent numerical check. Thus the load-bearing inequality s_b(N^2) <= N for N with at least three digits is not established by the printed proof.
[§3, Prop. 5(a)-(b)] The sentence "Clearly b) implies a)" is incorrect, since the right-hand side of b) contains N*b rather than N; in fact Proposition 5(a) is false as stated. A counterexample is N=[11]_2=3, for which s_b(N)=2 and 2s_b(N)=4>3. This lemma is cited in the proof of Proposition 7(a) to justify A>=0. The construction in Proposition 7 can be repaired by noting directly that the digit string defining N' has value at least its digit sum, but the manuscript's current proof relies on a false statement.

minor comments (5)

[§6, Case 2] The cases [22]_5, [22]_6, [22]_7, and [22]_8 are dismissed with "can be checked numerically" but the computations are not shown; a proof should either give the values of s_b(P^2) and P for these cases or replace the case split with a uniform argument.
[§5, Prop. 16] The displayed extra terms in Proposition 16 do not satisfy the defining equation. For N_k=[1(0)^k(0)^k1]_b and A_k=b^{2k}-2, one has s_b(N_k)=2, so A_k+s_b(N_k)=b^{2k}; its reversal is 1, giving b^{2k}+1, which is not N_k=b^{2k+1}+1 for k>=1. The intended sequence is likely [1(0)^k1]_b with A=b^{k+1}-2.
[§5, Prop. 23] In Proposition 23, for N_k=([1(0)^{k-1}1]_b)^2 and A_k=b^k-1, one has s_b(N_k)=4, so A_k+s_b(N_k)=b^k+3; multiplying by its reversal does not equal (b^k+1)^2 for general k. The correct extra term appears to be b^k-3, giving A_k+s_b(N_k)=b^k+1, whose square is N_k.
[§5, Prop. 14] The claim that a b-MRH number must be divisible by s_b(N)^2 is not generally true. For example, 90 is a 10-wMRH number with s_10(90)=9, and 9^2=81 does not divide 90. The final divisibility step in Proposition 14 therefore needs a different justification.
[§4, Prop. 7(a)] In the odd-length case of Proposition 7(a), the first displayed equality omits the middle digit in the summand; the printed equation should read [a_1...a_n(a_{n+1}/2)(0)^n]_b + [(a_{n+1}/2)a_n...a_1]_b = N. This is likely a typographical error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the main inclusions are proved by direct construction from digit-sum inequalities.

full rationale

The paper's central assertions (Propositions 7(a) and 17(a)) are proved directly. For palindromes N, Proposition 7(a) defines the additive extra term A from the string of digits of N and the digit sum s_b(N), then verifies equation (1) by base-b digit manipulation; nonnegativity of A follows from Proposition 5(a), an elementary bound proved independently in Section 3. For squares of palindromes, Proposition 17(a) defines A = P - s_b(P^2) and verifies equation (2); the required inequality s_b(P^2) ≤ P is Proposition 5(c) for palindromes with at least three digits, with the two-digit cases handled explicitly in Section 6. None of these steps assumes the target result or fits a parameter to the conclusion: the extra term A is given in closed form, and the digit-sum bounds are independent of the wARH and wMRH definitions. The paper cites earlier work by the same author for infinitude statements, inclusion of b-ARH and b-MRH numbers, and existence of palindromic multiples; those citations are not load-bearing for the two main inclusion theorems, and even if one set them aside the direct constructions stand. Finite checks left as 'can be checked numerically' are not circular, because they are finite verifications rather than use of the target conclusion. A possible mathematical gap in the Case 3 inequality (13) of Proposition 17(a) would be a correctness concern, not a circularity concern, and does not change this verdict.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

Most of the paper is elementary and self-contained, but it imports several results from the same author's earlier papers as black boxes, and it leaves a few finite checks to the reader. No free parameters or invented entities are introduced.

assumptions (5)

domain assumption For any integer N and base b, there exist infinitely many integers M such that N*M is a palindrome, cited as [8, Theorem 5].
Invoked in Corollaries 10 and 20 to show every integer divides a b-wARH and a b-wMRH number; not proved in this paper.
domain assumption Correctness of the prior theorems on ARH and MRH numbers from [6] and [8], specifically Theorems 13, 15, 26, and 35.
Used in Theorem 24, the proof of Theorem 36, Remark 8, and elsewhere; these are published results but not reproduced here.
standard math A base-b integer is divisible by b-1 if and only if its digit sum is divisible by b-1.
Used in the proof of Proposition 14 to distinguish wARH from MRH numbers via divisibility by (b-1)^2.
standard math Standard digit-sum bounds, such as s_b(N) at most (b-1) times the number of digits of N, and the reversal properties used in Sections 3 and 9 through 12.
Background for all the counting inequalities in Theorems 38 through 43; elementary but unstated in detail.
domain assumption The finite numerical checks omitted in Proposition 17 and Theorem 36 are correct.
The paper says 'can be checked numerically' for small bases and digits without listing computations; the class-inclusion proofs depend on these checks.

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Pith. "Pith review of About Some Relatives of Palindromes." pith.science (2026). https://pith.science/paper/ZOIJPWGZ

@misc{pith2026190800713,
  author       = {Pith},
  title        = {Pith review of: About Some Relatives of Palindromes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZOIJPWGZ}},
  note         = {Machine review of arXiv:1908.00713}
}

abstract

We introduce two new classes of integers. The first class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of digits of $N$, added to the reversal of the sum, gives $N$. The second class consists of numbers $N$ for which there exists at least one nonnegative integer $A$, such that the sum of $A$ and the sum of the digits of $N$, multiplied by the reversal of the sum, gives $N$. All palindromes that either have an even number of digits or an odd number of digits and the middle digit even belong to the first class, and all squares of palindromes with at least two digits belong to the second class. These classes contain and are strictly larger than the classes of $b$-ARH numbers, respectively $b$-MRH numbers introduced in Ni\c tic\u a \cite{N1}.

Figures

Figures reproduced from arXiv: 1908.00713 by the authors.

**Figure 1.** We observe that addition and multiplication of integers are independent of the numeration base. The operation of taking the reversal is not. Let sb(N) denote the sum of the base b digits of an integer N. Definition 2. A positive integer N written in base b is called b-weak Ramanujan-Hardy number, or simply b-wARH number, if there exists an integer A ≥ 0, called additive extra term, such that N = A + sb(N) + (A + sb(… view at source ↗

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Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

C. N. Cooper and R. E. Kennedy, On consecutive Niven numbers, Fibonacci Quart., 21 (1993), 146–151

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[2]

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[4]

Fredricksen, E

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High degree $b$-Niven numbers

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[9]

N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences , http://oeis.org. 2010 Mathematics Subject Classiﬁcation : Primary 11B83; Secondary 11B99. Keywords: base, b-Niven number, reversal, additive b-Ramanujan-Hardy number, multiplica- tive b-Ramanujan-Hardy number, hi...

2010

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Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

C. N. Cooper and R. E. Kennedy, On consecutive Niven numbers, Fibonacci Quart., 21 (1993), 146–151

work page 1993

[2] [2]

J. M. De Koninck and N. Doyon, Large and small gaps between con secutive Niven numbers, J. Integer Seq. , 6 (2003), Article 03.2.5

work page 2003

[3] [3]

H. G. Grundman, Sequences of consecutive Niven numbers, Fibonacci Quart., 32 (1994), 174–175

work page 1994

[4] [4]

Fredricksen, E

H. Fredricksen, E. J. Iona¸ scu, F. Luca, and P. St˘ anic˘ a, Re marks on a sequence of minimal Niven numbers, In: Sequences, Subsequences, and Consequences, Lec. Notes in Comp. Sci., Vol 4893. Springer, 2007, pp. 162–168

work page 2007

[5] [5]

G. H. Hardy, Ramanujan: Three Lectures on Subjects Suggested by his Life and Work , Chelsea, 1999

work page 1999

[6] [6]

Nit ¸ic˘ a, About some relatives of the taxicab number,J

V. Nit ¸ic˘ a, About some relatives of the taxicab number,J. of Int. Seq. , 21 (2018), Article 18.9.4

work page 2018

[7] [7]

High degree $b$-Niven numbers

V. Nit ¸ic˘ a, High degree b-Niven numbers, preprint, 2018, http://arxiv.org/abs/1807.02573

work page Pith review arXiv 2018

[8] [8]

Nit ¸ic˘ a, Inﬁnite sets ofb-additive and b-multiplicative Ramanujan-Hardy numbers, J

V. Nit ¸ic˘ a, Inﬁnite sets ofb-additive and b-multiplicative Ramanujan-Hardy numbers, J. of Int. Seq. , 22 (2019), Article 19.4.3

work page 2019