Exact termination of one seventh
in BASE 35

Philip John Jatzek and STORM (ChatGPT)
November 12, 2026

Our experiment confirmed that 1/7, which repeats forever in decimal, produces an exact zero remainder in our BASE 35 generator. After emitting 5, the generator emits zeros forever.

The calculation

The decimal expansion of 1/7 is 0.142857142857… . We multiply the exact starting value by BASE 35, emit the integer part, and retain the fractional remainder.

35 × (1/7) = 5

The integer part is 5 and the fractional remainder is exactly 0. Every later step multiplies zero by 35 and again produces zero.

Emitted sequence: 5, 0, 0, 0, 0, …

Two verification checks

Exact fraction arithmetic and a separate integer long division calculation each produced 5 followed by 999 zeros over 1,000 steps. Both ended with remainder exactly zero. The identity 35 × 0 = 0 proves that the zero tail continues forever, beyond the tested steps.

What the result means

The same number has a terminating representation in base 35: the base-35 numeral 0.5 represents 5/35, which equals 1/7. The value of 1/7 remains unchanged, and its decimal expansion still repeats.

This demonstrates a standard base-conversion property within our generator. BASE 35 is not unique: every positive integer BASE divisible by 7 produces an immediate zero remainder for the starting value 1/7.

Our documented result is an exact termination of the generator’s remainder for TV = 1/7 and BASE = 35. It is not a claim that repeating decimals have been eliminated in general.

REPORT NOV 12 2026 0425 PACIFIC

Near integer values of accumulated pi

Philip John Jatzek and STORM (ChatGPT)
November 12, 2026

We examined how close an integer multiple of pi can come to a whole number. Within 1 ≤ n ≤ 9,999,999,999,999, the closest approach occurs at n = 2,851,718,461,558. Its distance from the nearest integer is about 2.20198121 × 10⁻¹⁴. We also extended the calculation to larger bounds, shown on page 2.

The corrected model

Here n is a positive integer. PI TOTAL contains only n copies of pi. There is no extra addition of pi and no further multiplication by pi. E(n) measures the distance to the nearest whole number; zero would mean an exact whole-number result. This report replaces the earlier accumulated-pi-squared report for this intended model.

Verified results for the requested ranges

For n = 364,913, E(n) ≈ 5.877799729e-7. The product lies below 1,146,408.

For n = 2,851,718,461,558, E(n) ≈ 2.201981211e-14. The product lies above 8,958,937,768,937.

Closest product in the larger requested range

The fractional part is approximately 0.0000000000000220198121. The product is extraordinarily close to a whole number, but is not equal to it. The displayed decimal is an approximation.

Extending the search to larger bounds

The same method can search much larger ranges without enumerating every integer. The table includes the original larger range and three further bounds calculated for this report.

A larger bound can leave the best candidate unchanged until another record-setting denominator is reached. The errors above are absolute distances, whether the product falls above or below its nearest integer.

Continued fraction formulas

Write pi as a simple continued fraction. Its convergents pₖ/qₖ provide the record candidates for minimizing |qπ − p|, the best-approximation problem of the second kind.

Initialize p₋₂ = 0, p₋₁ = 1, q₋₂ = 1 and q₋₁ = 0. Generate convergents until the next denominator exceeds the chosen bound. The last eligible convergent supplies the best n = qₖ, with nearest integer pₖ and error |qₖπ − pₖ|. This is a mathematical search across the range, not a brute-force scan of trillions of values.

Verification and meaning

We repeated the continued-fraction calculations at 200 and 400 decimal digits. Both calculations agreed on every table entry and on the reported product digits. This is internal numerical verification, not outside replication or a formal interval certificate. The accompanying code and JSON retain the calculation and its results.

No nonzero integer multiple of pi is an exact whole number: if nπ = m, then π = m/n would be rational. Nevertheless, integer multiples can approach whole numbers arbitrarily closely. Continued fractions provide an established method for finding those approximations.

The finding is a reproducible near-integer approximation within stated bounds. It does not establish that pi terminates or becomes rational, and it is not a claim of mathematical novelty.

REPORT NOV 12 0451 PACIFIC

PI TOTAL search through
99999 digit multipliers

Philip John Jatzek and STORM (ChatGPT)
November 12, 2026

Full numerical report and conclusion

Main finding

We searched for an integer multiplier n that makes PI TOTAL = nπ as close as possible to a whole number, for 1 ≤ n ≤ 10⁹⁹⁹⁹⁹ − 1. The continued-fraction calculation selected a 99,999-digit multiplier. Its product falls just below a whole number, with a remaining distance of approximately:

The distance δ has 99,999 zeros immediately after the decimal point before its first nonzero digit. The product itself approaches the next whole number from below, so its fractional part contains a long initial string of 9s. The zeros belong to the distance from that whole number.

The precise question

n is a positive integer. PIT or PI TOTAL means n copies of pi only. No extra pi is added after forming nπ, and the result is not multiplied by pi again. This is the corrected model requested by Philip John Jatzek.

Result at a glance

Results across increasing bounds

The table summarizes the recorded best distances from the smaller searches through the current run. Each bound refers to integer multipliers, not the number of pi digits emitted by a recurrence.

Distances are rounded in this table. The zero count is the count immediately following the decimal point in the absolute distance before its first nonzero digit. A value c × 10⁻ᵏ, with 1 ≤ c < 10, has k − 1 such zeros.

What the decimal notation means

m is the nearest whole number. In the current result, δ is about 4.57 × 10⁻¹⁰⁰⁰⁰⁰. Consequently, rounding nπ to 99,999 places after the decimal point gives m followed by zero decimal places. That rounded display is not the exact value: the original product is still smaller than m.

The rational approximation m/n also approaches pi. Its error is δ/n, which differs from the search objective δ. This study minimized |nπ − m|; it did not claim that the selected rational is optimal under every other error definition.

Does adding digits always add zeros

No. The best distance improves at record-setting multipliers. Between those records, increasing the allowed bound can leave the best candidate unchanged. Larger bounds allow progressively smaller distances, but no universal one-zero-per-digit law follows from this table.

Method and verification

Continued fractions identify the candidates for the best approximations of the second kind. These minimize |qπ − p| as the allowed denominator q grows. Taking n = q turns that established approximation problem into our PI TOTAL question [1, 2].

Initialize p(−2) = 0, p(−1) = 1, q(−2) = 1 and q(−1) = 0. Generate convergents until the next denominator exceeds N. The last eligible denominator supplies the best candidate n under this objective, provided the continued-fraction coefficients have been computed correctly.

How the program computed the candidates

For each precision, Python mpmath computed pi. The program formed a long rational approximation floor(pi × 10ˢ)/10ˢ, where s equals the working precision minus 20. Exact integer Euclidean division then generated that rational approximation’s continued fraction. It retained convergents until the first denominator above the bound, and evaluated the selected error using the high-precision pi value.

The count includes the initial convergent. Timings describe these particular runs, not a performance guarantee. A 100,000-digit next denominator necessarily exceeds 10⁹⁹⁹⁹⁹ − 1.

Scope of confidence

Doubling precision while obtaining the identical candidate and reported distance is strong numerical evidence against a precision artifact. The integer continued-fraction recurrences themselves use exact arithmetic. However, pi was numerically approximated: these runs did not use directed interval bounds to certify every coefficient and the final error. The report therefore describes a numerically verified optimum under the standard theorem, not a formally certified computation.

The next verification step would use rigorous lower and upper bounds for pi, confirm a common continued-fraction prefix through the stopping point, and enclose nπ − m in an interval that excludes zero. Independent reproduction would provide an additional check.

STORM conclusion

The experiment succeeded at finding an exceptionally close approach to a whole number. It did not nullify the exact fractional part of pi. The useful result is a reproducible calculation showing how the chosen integer multiplier can make the distance extremely small, together with the full numbers and code needed to inspect that result.

This behavior is expected from established Diophantine approximation and continued-fraction theory. The calculation is not evidence that pi is rational, that its decimal expansion terminates, or that this is a previously unknown mathematical phenomenon. Its value is as a documented numerical experiment and a demonstration of the difference between exact equality and finite-precision agreement.

Why exact cancellation cannot occur

For integers n > 0 and m, the right-hand side is rational, while pi is irrational. Therefore E(n) cannot equal zero for any tested or future positive integer multiplier. Yet smaller nonzero errors can be found at larger bounds. Approaching zero is consistent with never reaching it.

Identifying the full numbers

The full multiplier and nearest integer are stored as decimal strings in RESULTS.json inside JATZEK_PI_TOTAL_99999_Digit_Search.zip. The report uses short identifiers below rather than printing about 200,000 digits. A SHA-256 hash identifies each complete string; hashes use UTF-8 digits with no commas, spaces or trailing newline.

Multiplier n — 99999 digits

First 40: 7593373768264519221862240898162642632301
Last 40: 1925100052898299962960384408505056085286

SHA-256: f86d6d98655d0a1c6dc4760b9995f275cc552a5b6f28b5fc19fa58cb7f10bf7b

Nearest whole number m — 100000 digits

First 40: 2385528724634125865362170954681497582265
Last 40: 4148310001155273125247831932997887927625

SHA-256: a7e1ccd11c2c07eb43d79320d8425101f2e23f2df855f8580beb40d4904a3913

Reproduction and references

Extract the calculation package and run python3 search.py with mpmath installed. The script requires Python support for large integer strings and disables the default digit conversion limit. It writes RESULTS.json and FINDINGS.md. The data and program accompany this report as a separate download.

[1] Gautam Gopal Krishnan. Continued Fractions. Cornell University, August 22, 2016, section 4.2. https://pi.math.cornell.edu/~gautam/ContinuedFractions.pdf

[2] Ian Short. Ford circles, continued fractions, and best approximation of the second kind. 2009. https://arxiv.org/abs/0912.1997

REPORT NOV 12 0451 PACIFIC