Emergence of New Infinite Sequences from Differences of Squares Organized by Maximal Delta
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Rui Ferreira
Feb 20, 2026, 7:06:27 PMFeb 20
to SeqFan
Hello SeqFan,
Abstract:
I present a systematic method for generating integer sequences derived from all numbers of the form x = b^2 - a^2 for non-negative integers a < b. By organizing these numbers according to the largest difference y = b-a for each x, we observe columns of sequences that exhibit a predictable structure: a finite irregular prefix followed by an infinite, stable quadratic tail. The stabilization start points follow the OEIS sequence A080827 (the rounded-up staircase), and the tail for column k follows the formula x = n(n + 2*(k-1)). These sequences are emergent from integer factorization and are not heuristic constructions, representing potential new entries for the OEIS.
1. Motivation:
The differences of squares, b^2 - a^2 = (b-a)(b+a), have long been studied in number theory. By examining these differences in an organized manner — specifically, by tracking the maximum delta y = b-a associated with each x — we aim to detect structured sequences hidden within the set of natural numbers.
2. Methodology:
1. Compute differences of squares:
For integers 0 ≤ a < b ≤ n_end, compute x = b^2 - a^2.
2. Track maximal delta:
For each x, retain the largest y = b-a that produces it. This yields a mapping x → y_max.
3. Group by delta:
Organize all x values into “columns” by y (the maximal delta). Column k contains all x where the maximal delta is y = k.
4. Analyze sequences:
- Quadratic fit for early columns: The first three columns (C1–C3) are quadratic and can be fitted explicitly.
- Empirical formula for columns k>3: For each column k, beyond a certain stabilization point n_start, all x-values follow the formula
x = n(n + 2(k-1)) for n ≥ n_start.
5. Identify stabilization:
The stabilization start n_start follows the OEIS A080827 staircase, giving a predictable first index for the stable tail.
6. Irregular prefix:
Before n_start, some x-values do not fit the formula. These finite prefixes represent “new sequences” before the stable infinite tail emerges.
3. Key Observations with examples:
- Column stabilization:
Each column eventually stabilizes at a predictable n_start (monotonic increasing with k).
- Tail formula:
After stabilization, the tail is exactly given by x = n(n + 2(k-1)), producing an infinite sequence.
- Irregular pre-stabilization examples values for columns 6 to 9 with new proposed sequences (out of infinite ones):
--------------------------------------------------
Column 6
--------------------------------------------------
Formula : x = n(n + 10)
Stabilizes at : n = 9 (A080827(3))
Run length : 772
Irregular x before stabilization: [13, 44, 88, 108, 176]
--------------------------------------------------
Column 7
--------------------------------------------------
Formula : x = n(n + 12)
Total values : 1993
Stabilizes at : n = 13 (A080827(4))
Match range : 13 → 1993
Run length : 1981
Irregular x before stabilization:
[17, 51, 52, 104, 132, 208, 312]
--------------------------------------------------
Column 8
--------------------------------------------------
Formula : x = n(n + 14)
Total values : 1992
Stabilizes at : n = 19 (A080827(5))
Match range : 19 → 1992
Run length : 1974
Irregular x before stabilization:
[19, 57, 68, 136, 156, 260, 272, 408, 512, 612]
--------------------------------------------------
Column 9
--------------------------------------------------
Formula : x = n(n + 16)
Total values : 1991
Stabilizes at : n = 25 (A080827(6))
Match range : 25 → 1991
Run length : 1967
Irregular x before stabilization:
[23, 69, 76, 115, 152, 204, 243, 300, 304, 456, 544, 648, 760, 1056]
--------------------------------------------------
- OEIS connection:
The staircase pattern of n_start exactly matches A080827:
[3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, ...]
- Predictability:
Given column k, one can predict:
- Stabilization start n_start (from A080827)
- Infinite tail formula x = n(n + 2(k-1))
- Number of irregular x before stabilization
Observation:
The irregular prefix remains constant when increasing n_end, while the tail continues indefinitely.
4. Interpretation:
- These sequences are emergent, fully determined by integer factorization and the maximal delta ordering.
- They are not heuristics or artificial formulas; the tail formula and the irregular prefixes arise naturally from the structure of differences of squares.
- The staircase stabilization is a manifestation of A080827, connecting triangular numbers to the emergence of these sequences.
5. Implications:
5.1. Each column 'Irregular x before stabilization' prefix represents a new potential unique OEIS sequence.
5.2. The approach provides a systematic way to discover and classify sequences derived from factorization patterns.
5.3. By varying n_end, one can catalog all irregular prefixes for each column, creating a reference for previously unknown sequences.
6. Conclusion:
This work reveals a new family of integer sequences emerging from the natural differences of squares, organized by maximal delta. The sequences combine finite irregular prefixes and infinite stable tails, with stabilization points dictated by OEIS A080827. These sequences are worthy of OEIS consideration, especially the finite prefixes before stabilization, which represent newly emergent integer sequences that have not yet been cataloged.
Author’s Note:
All sequences arise naturally from x = b^2 - a^2, organized by the largest delta y = b-a for each x. This method uncovers previously hidden structure in integer factorization and demonstrates predictable infinite sequence emergence.
Test code available at https://github.com/rf-iasys/OEIS/blob/main/OEIS_Irregular_prefix_before_stabilization_potential_new_OEIS_sequence_per_k.py
I welcome your comments.
With my best regards,
Rui Ferreira
Abstract:
I present a systematic method for generating integer sequences derived from all numbers of the form x = b^2 - a^2 for non-negative integers a < b. By organizing these numbers according to the largest difference y = b-a for each x, we observe columns of sequences that exhibit a predictable structure: a finite irregular prefix followed by an infinite, stable quadratic tail. The stabilization start points follow the OEIS sequence A080827 (the rounded-up staircase), and the tail for column k follows the formula x = n(n + 2*(k-1)). These sequences are emergent from integer factorization and are not heuristic constructions, representing potential new entries for the OEIS.
1. Motivation:
The differences of squares, b^2 - a^2 = (b-a)(b+a), have long been studied in number theory. By examining these differences in an organized manner — specifically, by tracking the maximum delta y = b-a associated with each x — we aim to detect structured sequences hidden within the set of natural numbers.
2. Methodology:
1. Compute differences of squares:
For integers 0 ≤ a < b ≤ n_end, compute x = b^2 - a^2.
2. Track maximal delta:
For each x, retain the largest y = b-a that produces it. This yields a mapping x → y_max.
3. Group by delta:
Organize all x values into “columns” by y (the maximal delta). Column k contains all x where the maximal delta is y = k.
4. Analyze sequences:
- Quadratic fit for early columns: The first three columns (C1–C3) are quadratic and can be fitted explicitly.
- Empirical formula for columns k>3: For each column k, beyond a certain stabilization point n_start, all x-values follow the formula
x = n(n + 2(k-1)) for n ≥ n_start.
5. Identify stabilization:
The stabilization start n_start follows the OEIS A080827 staircase, giving a predictable first index for the stable tail.
6. Irregular prefix:
Before n_start, some x-values do not fit the formula. These finite prefixes represent “new sequences” before the stable infinite tail emerges.
3. Key Observations with examples:
- Column stabilization:
Each column eventually stabilizes at a predictable n_start (monotonic increasing with k).
- Tail formula:
After stabilization, the tail is exactly given by x = n(n + 2(k-1)), producing an infinite sequence.
- Irregular pre-stabilization examples values for columns 6 to 9 with new proposed sequences (out of infinite ones):
--------------------------------------------------
Column 6
--------------------------------------------------
Formula : x = n(n + 10)
Stabilizes at : n = 9 (A080827(3))
Run length : 772
Irregular x before stabilization: [13, 44, 88, 108, 176]
--------------------------------------------------
Column 7
--------------------------------------------------
Formula : x = n(n + 12)
Total values : 1993
Stabilizes at : n = 13 (A080827(4))
Match range : 13 → 1993
Run length : 1981
Irregular x before stabilization:
[17, 51, 52, 104, 132, 208, 312]
--------------------------------------------------
Column 8
--------------------------------------------------
Formula : x = n(n + 14)
Total values : 1992
Stabilizes at : n = 19 (A080827(5))
Match range : 19 → 1992
Run length : 1974
Irregular x before stabilization:
[19, 57, 68, 136, 156, 260, 272, 408, 512, 612]
--------------------------------------------------
Column 9
--------------------------------------------------
Formula : x = n(n + 16)
Total values : 1991
Stabilizes at : n = 25 (A080827(6))
Match range : 25 → 1991
Run length : 1967
Irregular x before stabilization:
[23, 69, 76, 115, 152, 204, 243, 300, 304, 456, 544, 648, 760, 1056]
--------------------------------------------------
- OEIS connection:
The staircase pattern of n_start exactly matches A080827:
[3, 5, 9, 13, 19, 25, 33, 41, 51, 61, 73, 85, 99, 113, ...]
- Predictability:
Given column k, one can predict:
- Stabilization start n_start (from A080827)
- Infinite tail formula x = n(n + 2(k-1))
- Number of irregular x before stabilization
Observation:
The irregular prefix remains constant when increasing n_end, while the tail continues indefinitely.
4. Interpretation:
- These sequences are emergent, fully determined by integer factorization and the maximal delta ordering.
- They are not heuristics or artificial formulas; the tail formula and the irregular prefixes arise naturally from the structure of differences of squares.
- The staircase stabilization is a manifestation of A080827, connecting triangular numbers to the emergence of these sequences.
5. Implications:
5.1. Each column 'Irregular x before stabilization' prefix represents a new potential unique OEIS sequence.
5.2. The approach provides a systematic way to discover and classify sequences derived from factorization patterns.
5.3. By varying n_end, one can catalog all irregular prefixes for each column, creating a reference for previously unknown sequences.
6. Conclusion:
This work reveals a new family of integer sequences emerging from the natural differences of squares, organized by maximal delta. The sequences combine finite irregular prefixes and infinite stable tails, with stabilization points dictated by OEIS A080827. These sequences are worthy of OEIS consideration, especially the finite prefixes before stabilization, which represent newly emergent integer sequences that have not yet been cataloged.
Author’s Note:
All sequences arise naturally from x = b^2 - a^2, organized by the largest delta y = b-a for each x. This method uncovers previously hidden structure in integer factorization and demonstrates predictable infinite sequence emergence.
Test code available at https://github.com/rf-iasys/OEIS/blob/main/OEIS_Irregular_prefix_before_stabilization_potential_new_OEIS_sequence_per_k.py
I welcome your comments.
With my best regards,
Rui Ferreira
Gareth McCaughan
Feb 22, 2026, 8:31:07 AMFeb 22
to seq...@googlegroups.com
Unfortunately your description of what you are calculating doesn't match
your reported results, nor your actual code.
> 1. Compute differences of squares:
> For integers 0 ≤ a < b ≤ n_end, compute x = b^2 - a^2.
>
> 2. Track maximal delta:
> For each x, retain the largest y = b-a that produces it. This
> yields a mapping x → y_max.
>
> 3. Group by delta:
> Organize all x values into “columns” by y (the maximal delta).
> Column k contains all x where the maximal delta is y = k.
The following Python code implements 1-3.
> pairs = [(b*b-a*a,b-a) for a in range(100) for b in range(100) if a<b]
> by_x = {}
> for (x,y) in pairs: by_x.setdefault(x,[]).append(y)
> maxes = {x: max(ys) for (x,ys) in by_x.items()}
> by_y = {}
> for (x,y) in maxes.items(): by_y.setdefault(y,[]).append(x)
> for x in list(by_y.keys()): by_y[x].sort()
Then, according to your reported results, column 6 -- which should mean
either by_y[5] or by_y[6] depending on how you are counting -- is
supposed to contain 13, 44, 88, 108, 176 and then a lot of numbers of
the form n(n+10).
But by_y[5] begins: 25, 35, 45, 55, 65, 75, 85, 95, 115, 125, 145.
And by_y[6] begins: 36, 48, 60, 72, 84, 108, 132, 156, 204, 228.
Each of these sequences contains _exactly one_ of the numbers that you
say are in column 6.
I looked at your code. I think your function build_columns doesn't do
what you say it does. Its column k is _not_, as per your description,
"all x where the maximal delta is y=k", it's "the k'th-smallest x with
maximal delta y, for all y".
That seems a rather less natural thing to consider, and also -- for
reasons that will become apparent -- one where it's less surprising to
get a result of the form "it's basically n(n+k) with some early
complications". But, still, let's try to figure out what this actually
means. For a given y, the x with maximal delta y are the numbers of form
yz where y,z are the same-parity divisors of x closest to sqrt(x) with
y<z. That is: for a given y, they consist of y^2, y(y+2), y(y+4), etc.,
_except_ that we omit any of these that factor in a "closer" way.
So in some handwavy sense, "usually" we expect the 0-indexed k'th
smallest thing here to be y(y+2k), unless for some j<k we can rearrange
the factors of y and y+2j to produce a "closer" factorization of
y(y+2j). For instance, for y=5 you can see in the list shown above that
the entry 105=5.21 is absent, because 105 can also be written as 7.15;
for y=6 you can see that the entry 96=6.16 is absent, because 96 can
also be written as 8.12.
(I'm not sure how easy it is to get from the above handwaving to a proof
of Rui's empirical findings; I'll leave that to any others who might be
interested.)
--
g
your reported results, nor your actual code.
> 1. Compute differences of squares:
> For integers 0 ≤ a < b ≤ n_end, compute x = b^2 - a^2.
>
> 2. Track maximal delta:
> For each x, retain the largest y = b-a that produces it. This
> yields a mapping x → y_max.
>
> 3. Group by delta:
> Organize all x values into “columns” by y (the maximal delta).
> Column k contains all x where the maximal delta is y = k.
> pairs = [(b*b-a*a,b-a) for a in range(100) for b in range(100) if a<b]
> by_x = {}
> for (x,y) in pairs: by_x.setdefault(x,[]).append(y)
> maxes = {x: max(ys) for (x,ys) in by_x.items()}
> by_y = {}
> for (x,y) in maxes.items(): by_y.setdefault(y,[]).append(x)
> for x in list(by_y.keys()): by_y[x].sort()
Then, according to your reported results, column 6 -- which should mean
either by_y[5] or by_y[6] depending on how you are counting -- is
supposed to contain 13, 44, 88, 108, 176 and then a lot of numbers of
the form n(n+10).
But by_y[5] begins: 25, 35, 45, 55, 65, 75, 85, 95, 115, 125, 145.
And by_y[6] begins: 36, 48, 60, 72, 84, 108, 132, 156, 204, 228.
Each of these sequences contains _exactly one_ of the numbers that you
say are in column 6.
I looked at your code. I think your function build_columns doesn't do
what you say it does. Its column k is _not_, as per your description,
"all x where the maximal delta is y=k", it's "the k'th-smallest x with
maximal delta y, for all y".
That seems a rather less natural thing to consider, and also -- for
reasons that will become apparent -- one where it's less surprising to
get a result of the form "it's basically n(n+k) with some early
complications". But, still, let's try to figure out what this actually
means. For a given y, the x with maximal delta y are the numbers of form
yz where y,z are the same-parity divisors of x closest to sqrt(x) with
y<z. That is: for a given y, they consist of y^2, y(y+2), y(y+4), etc.,
_except_ that we omit any of these that factor in a "closer" way.
So in some handwavy sense, "usually" we expect the 0-indexed k'th
smallest thing here to be y(y+2k), unless for some j<k we can rearrange
the factors of y and y+2j to produce a "closer" factorization of
y(y+2j). For instance, for y=5 you can see in the list shown above that
the entry 105=5.21 is absent, because 105 can also be written as 7.15;
for y=6 you can see that the entry 96=6.16 is absent, because 96 can
also be written as 8.12.
(I'm not sure how easy it is to get from the above handwaving to a proof
of Rui's empirical findings; I'll leave that to any others who might be
interested.)
--
g
Robert McKone
Feb 22, 2026, 8:37:49 AMFeb 22
to seq...@googlegroups.com
Hey,
Gareth McCaughan, it is not worth your time to engage with Rui’s AI slop (it reads like AI slop, but if it is not, then that is not great for Rui).
I am being this blunt and honest because I truly believe Rui is throwing whatever is they can AI generate at a wall and see what sticks, and it angers me that this wastes everyone’s time in SeqFans. AI can be a great tool, but with how I see Rui using it, it is not being used correctly here and is just a slop-factory.
--
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Rui Ferreira
Feb 22, 2026, 8:44:23 AMFeb 22
to SeqFan
Gareth,
Thank you for your feedback, but I do not understand how could be so difficult to accept that these sequences are reproduced easily by the code I provide in the link and that none of these 'Irregular prefix' are documented in OEIS. The stabilization location " Stabilizes at :" where formula " Formula :" is given by A080827.
example:
--------------------------------------------------
Column 12
--------------------------------------------------
Formula : x = n(n + 22)
Stabilizes at : n = 51
Run length : 1530
Irregular prefix (potential new OEIS sequence):
[37, 111, 124, 155, 217, 248, 333, 348, 460, 496, 585, 588, 744, 828, 928, 1000, 1456, 1488, 1620, 1856, 2232, 2560, 2772, 3552, 3700]
Regards,
Rui
Thank you for your feedback, but I do not understand how could be so difficult to accept that these sequences are reproduced easily by the code I provide in the link and that none of these 'Irregular prefix' are documented in OEIS. The stabilization location " Stabilizes at :" where formula " Formula :" is given by A080827.
example:
--------------------------------------------------
Column 12
--------------------------------------------------
Formula : x = n(n + 22)
Stabilizes at : n = 51
Run length : 1530
Irregular prefix (potential new OEIS sequence):
[37, 111, 124, 155, 217, 248, 333, 348, 460, 496, 585, 588, 744, 828, 928, 1000, 1456, 1488, 1620, 1856, 2232, 2560, 2772, 3552, 3700]
Regards,
Rui
Gareth McCaughan
Feb 22, 2026, 9:06:52 AMFeb 22
to seq...@googlegroups.com
On 22/02/2026 13:37, Robert McKone wrote:
> Hey,
>
> Gareth McCaughan, it is not worth your time to engage with Rui’s AI
> slop (it reads like AI slop, but if it is not, then that is not great
> for Rui).
I have the same sense that it's AI-generated. It doesn't seem impossible
> Hey,
>
> Gareth McCaughan, it is not worth your time to engage with Rui’s AI
> slop (it reads like AI slop, but if it is not, then that is not great
> for Rui).
to me that there might be something that's interesting to someone in his
latest thing, though it would have been more interesting if the actual
process he described, rather than something more complicated and less
natural-feeling, had produced the results he says he got.
The question in my mind is whether Rui (or the system consisting of Rui
plus his LLM) is capable of learning what is and isn't worth posting.
We'll see.
--
g
Gareth McCaughan
Feb 22, 2026, 9:10:19 AMFeb 22
to seq...@googlegroups.com
On 22/02/2026 13:44, Rui Ferreira
wrote:
Gareth,
Thank you for your feedback, but I do not understand how could be so difficult to accept that these sequences are reproduced easily by the code I provide in the link and that none of these 'Irregular prefix' are documented in OEIS. The stabilization location " Stabilizes at :" where formula " Formula :" is given by A080827.
I am sure the sequences are reproduced by the code you provide.
However, the code you provide does not do what you say it does. Your description of it (or maybe your LLM's description of it) is wrong.
I am also sure you're correct that the "irregular prefixes" are
not presently in OEIS. Whether any given sequence _should_ be in
OEIS depends on whether it's mathematically interesting. If those
sequences had emerged from the process you _described_, that would
have been surprising and possibly interesting enough for them to
belong in OEIS. As it is, what you have to do to get them is more
complicated, more contrived, less interesting mathematically.
Maybe they still merit inclusion, maybe not, but they will
certainly not get in if there aren't accurate descriptions of what
they are beyond "my LLM wrote some code and this is what it
produced".
--
g
Robert McKone
Feb 22, 2026, 9:28:01 AMFeb 22
to seq...@googlegroups.com
Hello Rui I am replying publicly instead of privately to your private email,
SeqFan is about sequences. It is not about personalities, and it is not about “people like me”.
My objection is specific and technical. Your description of the construction did not match your code, and your reported column data did not match what the stated algorithm produces. Gareth demonstrated this clearly. That mismatch is the issue.
When I refer to AI slop, I mean material that is generated and posted without careful verification, internal consistency checks, or alignment between prose, code, and results. Whether it is produced by an AI tool or written manually is not the issue, the problem is the lack of rigor and the resulting noise in a technical forum.
If you want the work to be taken seriously, ensure that:
* The algorithm description matches the implementation.
* The implementation matches the reported data.
* Claims are supported by reproducible computation.
My objection is specific and technical. Your description of the construction did not match your code, and your reported column data did not match what the stated algorithm produces. Gareth demonstrated this clearly. That mismatch is the issue.
When I refer to AI slop, I mean material that is generated and posted without careful verification, internal consistency checks, or alignment between prose, code, and results. Whether it is produced by an AI tool or written manually is not the issue, the problem is the lack of rigor and the resulting noise in a technical forum.
If you want the work to be taken seriously, ensure that:
* The algorithm description matches the implementation.
* The implementation matches the reported data.
* Claims are supported by reproducible computation.
More bluntly, repeatedly posting work that does not survive even minimal consistency checks is not exploration, it is carelessness. Expecting others to debug conceptual and computational errors that should have been caught before posting shifts the cost of that carelessness onto the group. That is not collaboration, it is imposition. It is also selfish, SeqFan is a shared technical space. When material is posted without adequate checking, the burden of filtering, correcting, and repairing it falls on everyone else. That consumes time and attention that could have been spent on substantive mathematical discussion. Generating volume like you are doing with your AI slop and relying on others to supply the discipline is not a neutral act, it externalises the cost of your process onto the group.
That is a standards issue, not a personal one.
On Mon, 23 Feb 2026 at 00:49, Rui Ferreira (IASYS) <rui.fe...@iasys.eu> wrote:
Hi Robert,
So OEIS SeqFan is not about sequences ? It's about people like you ?
Regards,Rui
---------- Forwarded message ----------From: Robert McKone <r.p.m...@gmail.com>Date: February 22, 2026 at 1:37:36 pm -00:00Subject: Re: [SeqFan] Emergence of New Infinite Sequences from Differences of Squares Organized by Maximal DeltaTo: <seq...@googlegroups.com>
Hey,Gareth McCaughan, it is not worth your time to engage with Rui’s AI slop (it reads like AI slop, but if it is not, then that is not great for Rui).
I am being this blunt and honest because I truly believe Rui is throwing whatever is they can AI generate at a wall and see what sticks, and it angers me that this wastes everyone’s time in SeqFans. AI can be a great tool, but with how I see Rui using it, it is not being used correctly here and is just a slop-factory.On Mon, 23 Feb 2026 at 00:31, Gareth McCaughan <gareth.m...@pobox.com> wrote:
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--You received this message because you are subscribed to a topic in the Google Groups "SeqFan" group.To unsubscribe from this topic, visit https://groups.google.com/d/topic/seqfan/QyDflfoPoX0/unsubscribe.To unsubscribe from this group and all its topics, send an email to seqfan+un...@googlegroups.com.To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAN5y_DHyQNewTNRZ7%2B8ReOXLmiGh_C9%3Dq-%3DPdcAWqsSkp9ihSg%40mail.gmail.com.
Geoffrey Caveney
Feb 22, 2026, 9:32:13 AMFeb 22
to seq...@googlegroups.com
Thank you Robert. I agree with you.
Geoffrey
To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAN5y_DGK_qvLy17_aJPD1e8DLbG%2BvzaxbrFUXReST39-uHUJyw%40mail.gmail.com.
Jason Ausborn
Feb 22, 2026, 12:01:59 PMFeb 22
to seq...@googlegroups.com
Rui,
Take the positive from this discussion, the honesty shared and expectations are honorable.
Why? Math is unforgiving, it's honest, it's empirical, that's why we use it. A mistake can make the difference between life and death because it's used everywhere. Even for this message to reach you.
I'm not a true mathematician, but I've been factoring in my head to fall asleep for over 30 years. So many times, I thought I found a new discovery, a breakthrough, but several days into analyzing, I end up realizing my thoughts are still naive. Over and over and over... It's a loop!
Wanting to contribute, be a part of something, being the discoverer, it's natural for many of us. However, it's also important to accept the results. It drives us to be better, understand our work fully and become ready to defend it.
For myself, before submitting a claim as fact, I'm going to ask myself if it's ready, by thinking if it's not, it could kill someone. That will help improve the quality and accuracy of my work, whether it's AI assisted, or not.
Don't give up!
JA
To view this discussion visit https://groups.google.com/d/msgid/seqfan/CAFhKorUzHUNA68ezixgWbRzKtefK92OJRJrVwxWRr3%3DwMSgv2g%40mail.gmail.com.
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