TO: Simon Plouffe, Montréal, Quebec, Canada
FM: Bruce E. Camber
RE: Articles: Primes as sums of irrational numbers; homepage(s), ArXiv (10), Google Scholar, Wikipedia, YouTube
FileURL: https://81018.com/plouffe/
Eighth email: 28 October 2026
RE: Smart lookup request — one target, fixed class, fixed tolerance
Dear Prof. Dr. Simon Plouffe,
I’d like to run a disciplined version of your September inversion test, and I want the query fixed before your database answers, so the result can’t be shaped by what we expect to find.
The target: 0.6470879234 (10 digits, but only the first 6 should be considered meaningful — call it 0.647088).
The tolerance (fixed in advance, independent of any candidate): ±2 × 10⁻⁴. This is not tuned to any expression; it reflects the physical uncertainty behind the digits.
The class (fixed in advance): expressions of complexity no greater than a rational multiple p/q of one constant from {π, e, φ, √2, √3, ln 2, log₂ 3}, with p, q ≤ 12, or a sum of two such multiples of different constants; plus simple square/cube roots of small integers and log₂ of integers ≤ 8.
The questions:
- In your database, how many distinct expressions of this class fall within ±2 × 10⁻⁴ of 0.647088? Please list them all — I’d rather see the crowd than the champion.
- As a control: applying the same class and tolerance to, say, five random 6-digit targets of your choosing near the same range, how many hits does each get? This calibrates whether my target’s neighborhood is unusually crowded or unusually empty.
- If you widen the class one step (products of two constants, or a second level of sums), does the hit count near my target grow the way the control targets’ counts grow, or differently?
I’m deliberately not telling you where the target comes from, or which expression — if any — I might expect. That is the point: I’d like your answer to be blind to my hypothesis.
If a tolerance-window query isn’t something your inverter does directly, a “smart lookup” at 0.647088 with your own judgment about the neighborhood density would be a fair substitute, as would RIES or Maple’s identify run at 0.6470879234.
Thank you for your time — and I’m glad to share the full context and results afterward, once your count is in.
With respect,
Bruce
Seventh email: 20 October 2026
Dear Dr. Simon Plouffe,
Are the AI’s leading me astray or fine-tuning my work?
Here’s the latest: https://81018.com/base2_irrationals_wp/
Verdict? Thanks.
Sincerely,
Bruce
Sixth email: 9 September 2026
Dear Professor Plouffe,
Thank you for this extraordinarily generous and rapid response. You have answered my question more completely and honestly than I could have hoped.
Your output makes the situation clear: with only six significant digits, the number 0.687101 can be expressed by hundreds of formulas of comparable or lesser complexity to 2/3 + δ/2π. The list you sent is humbling — real roots of polynomials, hypergeometric functions, Riemann zeta zeros, combinations of π and γ — all producing the same six digits. Your shortest candidate, −39/10/π² + 17/5/π, is arguably simpler than our expression.
This does not falsify the conjecture, but it does precisely quantify the formula-selection concern that one of our reviewers (Perplexity) raised. You have now given us the honest answer: the numerical agreement at six digits carries very limited evidential weight on its own. The conjecture’s value, if it has any, must come from the geometric motivation for the specific formula — not from the numerical coincidence alone.
Your challenge is exactly right: more digits would help. The Planck 2018 measurement gives Ω_Λ = 0.6847 ± 0.0073 — eleven significant digits in the best current measurement would be 0.68470000000, but the uncertainty is in the third decimal place, so the physically meaningful precision is really only about three to four digits. Future measurements from DESI, Euclid, and the Rubin Observatory may eventually push to five or six digits of genuine precision in Ω_Λ.
Our formula gives exactly: 2/3 + (2π − 5arccos(1/3))/2π = 0.68710099973…
If future measurements converge on a value that matches these digits rather than simply the first three, the formula-selection problem diminishes. If they converge elsewhere, the conjecture is falsified. That is precisely the scientific situation.
With gratitude for your time and your honesty,
Bruce
Response: 9 September 2026 We requested and were granted permission to include this response with these materials.
Hello, Mr Camber.
At first glance, for example, here is an approximation of your number: -12/25 + 69/100 * 67^(1/8) = 0.6871009997346326… And that’s just the tip of the iceberg; here are some other candidates for your number, accurate to six decimal places: I can send you the output from my inverter directly; here I’ve only carried out a simple ‘lookup’ – if I try the ‘smart lookup’, I get hundreds of results.
The main issue is this: with just 6 digits, you can use my inverter (or others such as RIES or Maple’s ‘identify’ function) to find more. Or, if you like, if you give me your National Insurance number, a telephone number or the serial number of a plastic gadget: if it’s 15 digits or fewer, I can usually find a mathematical formula that produces those digits. In fact, the problem is filtering out all the false positives; the programme manages quite quickly to find a mathematical expression such as -39/10/π² + 17/5/π, which is fairly simple, and I have dozens like that: but then the real question is, which one is the right lead?
I cannot answer that for the simple reason that I do not know.
So, to answer your question: if you give me 41 decimal places of your constant, I can carry out very advanced tests; but with only 6 digits, anything is possible – including a whole host of false leads – and that’s where the problem lies.
Here is the output for 0.687100:
- 68710099482052143332747807306364688928654 a017 (89/55+159/59*37^(1/2))^(2/3)
- 68710099490570533657072053179139595207150 z004 n_th zero of Riemann Zeta,n=166302733
- 68710099558910648590484317126048203276626 a007 Real Root Of -15*x^4+649*x^3-465*x^2+62*x+476
- 68710099593390743925495333462836824695631 a017 -25/18+92/51*4^(2/3)*6^(1/3)
- 68710099606695938545256000238292761642181 h029 exp(Real Root of -1+x^15-x^14-x^13-x^12-x^10-x^9-x^8-x^7+x^5-x^4-x^3)
- 687100996244424974702670079216944427004112418 m023 3-1/2/Pi^2*2^(1/2)*3^(1/2)+12/Pi^2+1/3*Pi+3^(1/2)
- 68710099657464135107861058699831287598523 h028 log(hypergeom([1/17],[4/23,5/17,7/12],5/11))
- 68710099660138806723842081402144572557894 v030 sum((-1)^(n+1)/(2^n)/(n+3/10)/(n-1/18)/(n-2/3)/(n-5/8)/(n-5/24),n=0..infinity)
- 68710099681977097459727968441515987010732 m021 39/Pi^2-34/Pi
- 68710099693999151872385813682120089646903 a019 350963^(8/53)
- 68710099788449502232541625665445421168770 f002 hypergeom([-14/5],[7/12, 11/10, 5/2],2/11)
- 68710099821412103064247825292225574992782 a019 1411621^(56/59)
- 68710099847502707885195863752818037294737 m013 (1/6*ln(3)-2/3)/(4*ln(5)+3/5)
- 68710099866943089685957281623627294275439 m019 109/2*gamma^2+243/2*gamma-157
- 687100999080001741878013274429697509928795436 m023 -1/Pi^(1/2)/gamma-2/Pi/gamma^2-2*Pi^2*gamma^2-2*Pi^2/gamma^2
- 68710099909832155906296585446423360944627 f001 hypergeom([5/6,13/4,23/6],[-7/4,-2/3,-1/3],2/3)
- 68710099920188006347819133954103378168184 a016 (-20+497*91^(1/2))^(1/2)
- 68710099958293430928380962261653022412350 z004 n_th zero of Riemann Zeta,n=166302734
- 68710099973463262744381512707150460174008 a018 -48+69*67^(1/8)
- 68710099976699169650311482108834748001835 f001 hypergeom([2/3,5/3,13/6],[-5/6,7/2,15/4],7/12)
- 68710099992232834638429676671000074927781 r005 Im(z^2+c),c=-23/114+14/19*I,n=48
- 68710099999590943877708896570065898027125 h032 exp(-1/12*Pi)-ln(-7/5*Pi)-ln(-11/6*Pi)-1/Pi
So far, the shortest answer I have is -39/10/π² + 17/5/π
Can you get more digits of that value ? if it is based on a physical measurement then, I am afraid that those measurements are limited to 11 digits in the best case.
Best Regards,
Simon Plouffe
Fifth email: 9 September 2026
Dear Professor Plouffe,
You have spent your career asking exactly the question I need help with, and I am writing to ask it directly.
A geometric conjecture proposes that the dark energy fraction of the universe equals:
Ω_Λ = 2/3 + δ/2π = 0.687101…
where δ = 2π − 5arccos(1/3) is the Aristotle gap — the exact angular deficit when five regular tetrahedra share a common edge in three-dimensional Euclidean space. The value 0.687101 is within 0.33 sigma of the Planck 2018 measured dark energy fraction of 0.6847 ± 0.0073.
The conjecture uses no cosmological inputs and no free parameters. But one of its reviewers correctly identified the central weakness: the combination rule — adding 2/3 to δ/2π rather than subtracting, squaring, or otherwise combining them — was chosen with knowledge of the observational target. This makes the statistical weight of the 0.33 sigma agreement dependent on how many equally simple exact expressions for 0.687101 exist.
This is precisely the kind of question your Inverse Symbolic Calculator was built to answer.
My specific question: is 2/3 + (2π − 5arccos(1/3))/2π the unique simplest exact closed-form expression for 0.687101, or are there other natural geometric expressions of comparable simplicity that produce values equally close to the Planck 2018 measurement? If the expression is essentially unique among expressions of its complexity class, the numerical agreement carries more evidential weight. If there are many equally simple alternatives, it carries less.
I recognize this is an unusual request. But I believe it is exactly the kind of formula-selection audit that responsible presentation of a numerical conjecture requires — and your tools are the right ones for it.
The full paper: https://81018.com/geometric-conjecture-dark-energy-fraction/
With the greatest respect for your work
and thanks for your help in the past,
Bruce
Fourth email: 2 March 2026 (Updated)
Dear Prof. Dr. Simon Plouffe:
In March 2025 working with Grok, examining the role of pi (π) and the three inherent facets of pi (π), continuity-symmetry-harmony, I raised the question about the role of the other primary irrationals — e, φ, √2 — in stabilizing the sphere. We discovered that indeed each had a role. I asked, “Could they all be part of Notation-0 with all the Planck base units and all their dimensionless constants?”
Since 1999, I asked questions about the four natural hexagonal plates within the octahedron. The answer to that question was, “Yes.”
By July, that model came alive as a dynamic graphic. I was getting it ready for PASCOS-30 at IPPP of Durham University in Durham, England: https://81018.com/planck-polyhedral-core/. This week’s homepage returns to the emergence of necessary geometries — https://81018.com/notations-0-10/ — right from the start. I am getting nice support from the six major AI systems: https://81018.com/2026-state-of-the-universe-d/
Might you have a few comments?
Thank you.
Most sincerely,
Bruce
Third email: 18 December 2024
Dear Prof. Dr. Simon Plouffe,
Fantastic summary! If there are not even small groupings of patterns, what are those endless numbers telling us?
1. Continuity is the bedrock of the universe?
2. Circles and spheres never close the gap?
3. Infinity is profoundly in the finite… the finite has a leg of the infinite
Also, please advise me on this paragraph:
“The Planck base units manifest within a base-2 expansion. Stoney units or ISO–BIPM units will be used for comparative studies. It is a system of numbers of natural units where fundamental constants like the speed of light and the gravitational constant are set to dimensionless values of 1 effectively removing them from equations and making all derived Planck units dimensionless in relation to each other. These are eidetic numbering systems (See Simon Plouffe for more). Taken by itself, each number is defined by equations, ratios of foundational tensions and qualities foremost among them being space-time, mass-energy, and electromagnetism-gravity.”
Is it all just too idiosyncratic?
Cheers to you,
Bruce
Second email: 17 July 2022, 5:18 PM
Dear Prof. Dr. Simon Plouffe,
We are still alive-and-well and moving slowly ahead. Is this statement about your work okay? I said, “And, Simon Plouffe has through algorithmic programming identified over 215 million constants. There is nothing in the universe not touched by dimensionless constants.”
Also, I am sure you have seen the five-tetrahedral gap that Aristotle missed.
Have you ever seen a five-octahedral gap? It seems that we all missed it.
Combined, the two make a very sweet image: https://81018.com/15-2/
I am not sure if its analogue is to a gate in circuitry or to a Zen koan!
I propose that it is the first step in a geometry of the gaps, quantum geometry, squishy geometry, imperfect geometries or even perhaps the geometry of quantum fluctuations. Your comments?
Laughter is perfectly acceptable and understandable! Speaking of perfection: https://81018.com/perfection/
By the way, I have a page on our site about you and your work: https://81018.com/plouffe/ Any changes? Just say the word. Thanks so much.
Best wishes,
Bruce
Response: March 8, 2018
Professor Simon Plouffe responded and gave us permission to post his answer:
“There are many answers:
- Most of the numbers are irrationals.
- Every table has a name, from a to z with 3 digits, like a008, is the 8’th table of algebraic numbers.
- There is a category of rational numbers like q001, q002.
- It is widely believed that the number gamma (euler constant) is irrational but no proof of that exist. It is the case with most of the numbers in the tables, we have no proof that they are irrationals (but most likely to be). Proofs of irrationality are tough and difficult to make.
- In other words, we don’t know much about the very nature of many mathematical constants.
- It is widely believed that most of the real numbers are transcendental: https://en.wikipedia.org/wiki/Transcendental_number
First email: Monday, Feb 19, 2018 at 11:10 PM
Dear Professor Simon Plouffe:
Our work began in December 2011 within high school geometry classes where we followed the tetrahedron-octahedron, going within, by dividing the edges in half, deeper and deeper, 112 steps to the Planck scale and then we followed it out 90 steps by multiplying by 2, to the Observable Universe. We thought it was a good STEM tool. On further consideration, the first 67 notations to the CERN-scale began to intrigue us.
A few years later, we added Planck Time to our Planck Length chart, then two years later we added Planck Charge and Planck Mass. As we studied the numbers we began to think that we lived in an exponential universe and thought Euler might be pleased. Certainly the Hawking-Guth team were not. Here was a natural inflation that did not defy all logic. Then we began looking for alternatives to absolute space and time.
To say that we are a bit idiosyncratic captures some of the flavor of this work.
Now, we have discovered your work and we are celebrating. What marvelous things you have done and are doing. You’ll be teaching us a lot!
We wrote a small summary about Hilbert’s sense of the infinite and made our first reference to your 11.3 billion mathematical constants. It is below and part of our website: https://81018.com/finite-to-infinite/#3
Are all those 215 million non-ending, non-repeating numbers? I hope so. Thank you.
Most sincerely,
Bruce
****************
Bruce Camber
http://81018.com
Our reference to you is highlighted.
Excerpt from the homepage:
What is infinite? In 1925, the great mathematician, David Hilbert wrote, “We have already seen that the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to.” Many scholars would agree even today, but maybe Hilbert and those scholars are mistaken. There are many non-ending and non-repeating numbers such as pi, Euler’s equation (e), and all the other dimensionless constants. Aren’t these numbers evidence or a manifestation of the infinite within the finite?
Yes, I believe access to the infinite is found in the primary dimensionless constants where the number being generated does not end and does not repeat. There are 26-to-31 such numbers that have been associated by John Baez and Frank Wilczek-and-others to be necessarily part of the definition of the Standard Model of Particle Physics. There are over another 300 such numbers defined by the National Institute for Standards and Technology (NIST). All are dimensionless constants that seemingly never-end and never-repeat. And, then there is Simon Plouffe; he has identified, through algorithmic programming, 215 million mathematical constants (as of August 2017) which includes pi, Euler’s number, and more. This use of “never-ending, never-repeating” as the entry to the infinite will be challenged. If it can be defended, then there are more connections between the finite and infinite than David Hilbert and most scholars had ever anticipated. More…