REVIEW 1 major objections 6 minor 5 references
Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages
T0 review · 1 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read Oasis lengths for augmented happy functions are bounded, with exact maxima through base 20.
desk verdict A clean, modest sequel that completes the oasis picture for small bases; the math is sound but the exact maxima for b≤20 depend on an asserted search that should be shipped. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the $k$-mirage base $b$: a set of $k$ consecutive integers of the form $d = r - S_{[0,b]}(r)$, with $r$ ranging over non-negative integers having at most three base-$b$ digits. The paper shows that two fixed points lying in the same oasis must agree in every digit beyond the third (Theorems 5 and 6), so the augmenting constant of a fixed point is determined entirely by its last three digits. Using $c = a - S_{[0,b]}(a)$ (Equation (1)), every oasis is shown to produce a mirage, and every mirage consisting only of positive integers is an oasis (Lemma 9, Theorem 10). Searching all $0<r<b^3$ for the longest consecutive run of $d$-values therefore gives an upper bound on oasis length, and in the cases $2\le b\le 20$ the search also attains the bound.
What would settle it
Run an independent implementation of the algorithm for each even base $2\le b\le20$: compute and sort all $d=r-S_{[0,b]}(r)$ for $0<r<b^3$ and inspect the longest run of consecutive $d$ values; if any run is longer than the maximum reported in Theorem 11 ($2,6,5,8,8,8,8,8,8,9$ for $b=2,4,6,8,\dots,20$), the theorem is false.
Extended reading notes
Core claim
On its own terms, the paper establishes that a $k$-oasis base $b$ exists only for $k \le b^3/2 + b^2/2 - b$ when $b$ is even, while for odd $b$ every oasis has length $1$. It then proves the bound is attained in the small even cases by displaying the exact maximal lengths: $2$ for $b=2$, $6$ for $b=4$, $5$ for $b=6$, $8$ for each of $b=8,10,12,14,16,18$, and $9$ for $b=20$. Table 1 records, for each of these bases, the smallest example of a maximal-length oasis and the smallest fixed point for each augmenting constant in it.
Load-bearing premise
The claimed exact maxima rest on the finite computer search being complete and correct; a missed longer run of candidate values would make a listed maximum too small.
Editorial extensions
If this is right
- For any fixed base $b$, oasis length is finite: no matter how large the augmenting constants become, a block of more than the maximum number of consecutive fixed-point-bearing $c$ values cannot occur.
- In odd bases every oasis has length exactly $1$, so only even bases can exhibit nontrivial oases.
- For every even base $2\le b\le20$, a block of $c$ values of length $2$, $6$, $5$, $8$, or $9$ (according to base) can be found starting at the minimal example in Table 1, and any longer block would contain a gap.
- Whenever a $k$-oasis base $b$ exists, infinitely many $k$-oases exist, so the maximal examples found are not unique; they repeat at larger values of $c$.
- Computing the maximum oasis length reduces to a finite check over $r<b^3$; no search over all fixed points is needed.
Reading between the lines
- Inference: The same finite mirage search should determine exact maximal oasis lengths for even bases beyond $20$, with the practical limitation being the $O(b^3\log b)$ cost of enumerating and sorting all $r-S_{[0,b]}(r)$ values; the paper stops at $b=20$.
- Inference: The gap between the general upper bound $\sim b^3/2$ and the computed maxima (all $\le9$) suggests that the true maximal oasis length grows much more slowly than the bound, and the asymptotic growth rate is left open.
- Inference: The reduction to the last three digits is tied to the square in the digit-sum function; for sums of higher powers of digits, the analogous fixed-point sets would have a different finite cut-off, so oasis lengths for higher-power happy functions need not follow the same pattern.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines a k-oasis base b as a set of k consecutive nonnegative augmenting constants c for which the augmented generalized happy function S_{[c,b]} has at least one fixed point. It proves that every base b has an oasis, that odd bases have only oases of length 1, that even b ≥ 6 have 5-oases, that the existence of one k-oasis implies infinitely many, and that fixed points belonging to the same oasis agree in all but the three least significant digits. It derives a general bound k ≤ b^3/2 + b^2/2 − b and, via a reduction to a finite search over values d = r − S_{[0,b]}(r) with 0 < r < b^3, reports exact maximal oasis lengths for all even bases b ≤ 20. The lower-bound constructions are explicit and checkable in Table 1.
Significance. The theoretical development is sound, elementary, and clean: the reduction of oasis length to mirage length (Lemma 9 and Theorem 10) is a nice contribution, and the general bound in Theorem 7 is explicit. If the computational half is verified, Theorem 11 settles the maximal oasis lengths for all bases up to 20 and shows the general bound is nearly sharp. The paper's lower-bound examples are concrete and machine-verifiable by hand. The main weakness is that the exact maximality claims in Theorem 11 rest on an exhaustive computer search that is described but not shipped, so the upper-bound half is not independently reproducible from the manuscript.
major comments (1)
- [Section 3, Algorithm steps 1–4, and Theorem 11] The exact maximal lengths asserted in Theorem 11 for bases 2 through 20 depend entirely on the claim that the finite enumeration over 0 < r < b^3 was performed correctly and completely. The manuscript provides the algorithm but no code, no sorted d-values, no longest-run output, and no certificate. A missed longer consecutive run, including one involving negative d-values, would invalidate the stated upper bounds and hence the exactness of Theorem 11. Please provide the implementation and its output (for example, the sorted distinct d-values and the longest runs for each base), or explicitly restate Theorem 11 as a conditional-on-search result. This is the load-bearing gap in the paper's central claim.
minor comments (6)
- [Section 2, proof of Theorem 5] The proof contains a duplicated paragraph: the statement of the bound from [2, Theorem 4.2] appears twice, and in the second occurrence the subscript in "mb,ˆn" should read "mb,¯n". Please delete the duplicate and correct the typo.
- [Section 2, after Theorem 5] The example for optimality uses the notation "10(15) (16)"; consider writing it as (10 15)_16 or 1·16^2 + 15 to avoid confusion with multiplication.
- [Section 3, Algorithm step 1] The algorithm restricts to 0 < r < b^3 and excludes r = 0; this is harmless because r = 1 gives d = 0, but the equivalence should be stated explicitly so that the reader sees no d-value is lost.
- [Section 3, Theorem 10 proof] In the displayed equation following Eq. (2), the summation index is written as "∑_{j=3}^n" but should be "∑_{i=3}^n"; this is a typographical error that does not affect the derivation.
- [Table 1] The table lists "Smallest fixed points" for each maximal oasis but does not indicate which fixed point corresponds to which augmenting constant in the oasis; adding that correspondence, or at least stating the ordering of the oasis elements, would improve verifiability.
- [Section 3, discussion of Table 1] The text says the table provides the "minimal example" of a maximal length oasis for each even base, but no ordering or search for minimality is described; if "minimal" is intended as a claim, it needs justification, and if it merely means "an example found by the search," the wording should be relaxed.
Circularity Check
No circularity: the bounds and exact oasis lengths are derived from prior fixed-point lemmas and an explicit finite enumeration, not from the claims themselves.
full rationale
The paper's derivation chain is self-contained in the relevant sense. Theorem 7's upper bound follows from Theorem 6, which restricts which digits of fixed points may differ, together with cited digit-symmetry results [2, Theorem 2.1 and Lemma 2.2]; those prior results concern fixed points of augmented happy functions and do not include the oasis-length claims being proved. The mirage reduction in Theorem 10 is proved from Theorems 5 and 6, not assumed. The Section 3 algorithm enumerates d = r - S_{[0,b]}(r) for each 0 < r < b^3, sorts the values, and finds the longest consecutive run; Lemma 9 then converts a positive run into an oasis. No oasis length, bound, or maximal-length value is used as an input to this enumeration, and no parameter is fitted to the data being predicted. The finite exhaustive search underlying Theorem 11 is described but not shipped as code or data, so its correctness is a verifiability concern rather than a circularity concern. The paper does cite results from the authors' earlier Part I, and those citations are load-bearing in several proofs, but they are published, peer-reviewed mathematical results whose assumptions do not include the present target results; under the stated rules, that is independent support and does not raise the circularity score. No equation in the paper reduces to its own conclusion by construction.
Assumptions & free parameters
assumptions (4)
- domain assumption For odd b ≥ 2, if S[c,b] has a fixed point then c is even (Part I, Lemma 2.3).
- domain assumption If a is a fixed point of S[c,b] and a is a multiple of b, then a+1 is also a fixed point (Part I, Theorem 2.1).
- domain assumption If a fixed point has second rightmost digit d ≠ 0, the number obtained by replacing that digit with b-d is also a fixed point (Part I, Lemma 2.2).
- domain assumption For n ≥ 2, if S[c,b] has a fixed point of n+1 digits then m_{b,n} ≤ c ≤ M_{b,n} with explicit sharp bounds, and M_{b,n}+1 < m_{b,n+1} (Part I, Theorem 4.2 and Lemma 4.3).
Cite this review
Pith. "Pith review of Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages." pith.science (2026). https://pith.science/paper/OJWO7YC6
@misc{pith2026190802194,
author = {Pith},
title = {Pith review of: Fixed Points of Augmented Generalized Happy Functions II: Oases and Mirages},
year = {2026},
howpublished = {\url{https://pith.science/paper/OJWO7YC6}},
note = {Machine review of arXiv:1908.02194}
}
abstract
An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. For $b\geq 2$ and $k \in \mathbb{Z}^+$, a $k$-desert base $b$ is a set of $k$ consecutive non-negative integers $c$ for each of which $S_{[c,b]}$ has no fixed points. In this paper, we examine a complementary notion, a $k$-oasis base $b$, which we define to be a set of $k$ consecutive non-negative integers $c$ for each of which $S_{[c,b]}$ has a fixed point. In particular, after proving some basic properties of oases base $b$, we compute bounds on the lengths of oases base $b$ and compute the minimal examples of maximal length oases base $b$ for small values of $b$.
Reference graph
Works this paper leans on
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[1]
B. Baker Swart, K. Beck, S. Crook, C. Eubanks-Turner, H. Gr undman, M. Mei, and L. Zack, Augmented generalized happy functions, Rocky Mountain J. Math. , 47 (2017), 403–417
work page 2017
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[2]
B. Baker Swart, K. Beck, S. Crook, C. Eubanks-Turner, H. Gr undman, M. Mei, and L. Zack, Fixed points of augmented generalized happy functions, Rocky Mountain J. Math., 48 (2018), 47–58
work page 2018
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[3]
H. G. Grundman and E. A. Teeple, Generalized happy numbers, Fibonacci Quart., 39 (2001), 462–466
work page 2001
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[4]
Guy, Unsolved Problems in Number Theory, third ed., Springer-Verlag, 2004
Richard K. Guy, Unsolved Problems in Number Theory, third ed., Springer-Verlag, 2004
work page 2004
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[5]
AMS 2010 Mathematics Subject Classification: Primary 11A63
Ross Honsberger, Ingenuity in Mathematics , New Mathematical Library, 23, Random House, Inc., 1970. AMS 2010 Mathematics Subject Classification: Primary 11A63. Keywords: happy number, fixed point, iteration. 9
work page 1970
Reviewed August 14, 2026 · model on record in the stance chip above.
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