Bonus Comparative Challenge

37 and 73 versus the secular controls 25 and 52

Formal status: TBP-121. This entire page is one composite comparative requirement within the current 290 documented requirements—not a collection of additional independent requirements. The core observations below are fixed and disclosed together. Cross-references to Patterns 1–100 are not counted again, and the curiosities and experimental digit searches are explicitly uncounted.

What a challenger must do

A challenger is not required to reproduce the exact prime, geometric, or textual properties of 37 and 73. That would be impossible for many properties because 25 and 52 are composite. Instead, the challenger may submit a comparably strong collection of independently verifiable properties involving 25 and 52.

The three-judge panel will compare the two collections for mathematical naturalness, number of independent results, reproducibility, use of established definitions, and freedom from arbitrary after-the-fact choices. As required by the published rules, acceptance requires a unanimous decision.

Published comparison rules

Every claim must be mathematically correct and supported by a complete calculation, reproducible program, or reliable source.
Direct consequences and equivalent reformulations of one fact count only once.
Every spelling, base change, reversal, concatenation, omitted digit, search range, starting position, and interval must be disclosed.
A rule developed for one pair must be applied symmetrically to the other pair.
Arbitrary constants or operations selected only to force a desired result receive little or no weight.
Properties common to most integers do not establish that either pair is unusually significant.
The approved three-judge panel decides comparability under these rules; it may reject duplicate, weak, ambiguous, or unreproducible submissions, and its decision must be unanimous.

Core comparative observations

1. 37 and non-supersingular primes Specialized

In monstrous moonshine theory, 37 is the smallest prime that is not supersingular. Every prime p at or above 73 is also non-supersingular. These are two parts of one mathematical fact, not two independent observations.

Background: 37 (number) and supersingular primes in moonshine theory.

2. Centered-hexagon and star geometry Strong

37 is both a centered hexagonal number and a star number, while 73 is the immediately following star number:

H₄ = 3(4)(3)+1 = 37

S₃ = 6(3)(2)+1 = 37 and S₄ = 6(4)(3)+1 = 73

The 37-point fourth centered hexagon is the exact central core of the 73-point fourth star array. The 37-point and 73-point stars are consecutive star-number arrays. Neither 25 nor 52 belongs to either sequence.

View the geometric illustration.

3. A reversal-and-doubling prime pair Strong

reverse(73)=37

(73+1)/2=37 or 73=2(37)-1

A computer search found no other prime pair below one billion satisfying both conditions. The control pair reverses in decimal notation, but neither control number is prime and (52+1)/2 is 26.5.

View reversibleprimes.java.

4. Three levels of numerical reflection Strong

The values reverse: 37 ↔ 73.
The prime positions reverse: 37 is the 12th prime and 73 is the 21st prime.
The squared positions reverse: 12²=144 and 21²=441.

Because 25 and 52 are composite, they have no positions in the prime sequence.

5. Odd positions and Fibonacci divisibility Strong

Number	Odd-number position	First Fibonacci divisibility index
37	19	19
73	37	37

Thus 37 is the 19th odd number and first divides a Fibonacci number at index 19; 73 is the 37th odd number and first divides a Fibonacci number at index 37. Both divide Fibonacci numbers whose indices are multiples of 703=19×37.

The controls have ordinary Fibonacci ranks—25 at index 25 and 52 at index 42—but do not reproduce the linked odd-position progression. fib.java | odd-number table

6. Nested triangular arrays and “and the earth” Strong

T₃₇=37×38/2=703

T₇₃=73×74/2=2701=37×73

Because 73=2(37)−1, the 37-point-per-side triangular array fits centrally and inverted within the 73-point-per-side array.

The final two Hebrew words of Genesis 1:1, וְאֵת הָאָרֶץ (ve'et ha'aretz, “and the earth”), have standard value:

407+296=703=T₃₇

The circles in the illustration symbolically represent the Earth; their shape is illustrative rather than part of the calculation. See also Pattern 72.

7. A decimal construction leads to binary 73 Interpretive

1+2+3+4+5+6+7+9=37

12,345,679×3=37,037,037=37×1,001,001

The same digit string 1001001, reinterpreted in base 2, is the palindromic binary representation of 73:

1001001₂=64+8+1=73

The omission of 8, multiplication by 3, decimal factorization, and base change are all disclosed. The controls have non-palindromic binary forms: 25=11001₂ and 52=110100₂.

8. Repeating decimals of sevenths Verified

1/7=0.142857

1+4+2+8+5+7=27=3³

142857=27×5291=37×3861

For every integer not divisible by 7, the fractional part of its quotient by 7 has one of the six cyclic rotations of 142857. Every rotation has digit sum 27 and is divisible by both 27 and 37. The earlier example 3/7=0.428571 is one case of this general result.

View the finite Java demonstration. The universal result follows from the remainder classes modulo 7.

9. The 27×37=999 decimal relationships Strong

27×37=999

1/37=27/999=0.027 and 1/27=37/999=0.037

The first 1,010 positive multiples of 999 all have digit sums divisible by 27. Of these, 1,009 have digit sum 27; 999×1001=999999 has digit sum 54. The next multiple, 999×1011=1009989, has digit sum 36, so the run ends.

The identical control constructions fail immediately: digitSum(27×25)=18 and digitSum(27×52)=9. View special.java.

10. Factorial digit sums and 153 fish Verified

digitSum(37!)=153

digitSum(73!)=315

315 is obtained by moving the final digit of 153 to the front. John 21:11 records 153 large fish in the miraculous catch made by the disciples after following Jesus’ instruction.

The control results are digitSum(25!)=72 and digitSum(52!)=279.

11. “Jesus” and the Mispar Shemi value of Genesis 1:1 Interpretive

JESUS=10+5+19+21+19=74=37+37

74²=5476

Using the explicitly listed Mispar Shemi letter-name spellings and values, the 28 Hebrew letters of Genesis 1:1 also total 5476. Mispar Shemi spellings can vary, so the supporting page must remain part of the disclosure.

View the complete calculation.

12. “John Baptist” in Greek isopsephy Textual

ΙΩΑΝΝΗΣ=1119 and ΒΑΠΤΙΣΤΗΣ=1101

1119+1101=2220=60×37

The calculation uses the Greek name and title without the definite article ὁ. Including the article would produce a different total. Neither 25 nor 52 divides 2220.

13. Concatenated Genesis letter values modulo 777 Interpretive

Concatenating the standard values of the 28 Hebrew letters in textual order, without padding, produces:

N=2200130010400220011305104014005300401040614005120090

N mod 777=307

Since 777=3×7×37, the remainder visually displays 3 and 7 separated by zero. The arithmetic is exact; the visual meaning assigned to 307 is number play. The same integer gives remainders 15 modulo 25 and 34 modulo 52.

14. The first 37 prime squares Strong

Σ_k=1..37 p_k²=263736=37×7128

Here p_k is the kth prime, so the sum runs from 2² through 157². The identical tests fail for the controls: the first-25-prime-square sum leaves remainder 21 modulo 25, and the first-52-prime-square sum leaves remainder 3 modulo 52.

View prime.java.

15. Two disclosed numerical associations Lower strength

Temperatures and mercury. 37°C is traditionally cited as normal human body temperature, while 73°F is a comfortable room temperature. Mercury, historically used in thermometers, is element 80 and has 80 protons. Taking the leading digit from three descriptions involving 80 produces 888, the Greek isopsephy value of Ἰησοῦς. The three descriptions of 80 are not independent, and dropping the zeros is explicitly disclosed.

Repeated 7.

7×7×77=7³×11=3773=37|73

The symbol | denotes decimal concatenation in this display, not multiplication or divisibility.

Related Patterns 1–100—not counted again

The following material appeared on the old Bonus page but is already documented as part of the first 100 patterns:

777, 888, 999, and 4995: Patterns 45–48.
The Hebrew values associated with “forever” and “truth”: Pattern 62.
The Hebrew ordinal value 37 and standard value 73 of “wisdom”: Pattern 52.
2701+1072=3773: Pattern 71.
2701=37×73 and its triangular form: Pattern 72.
26²+45²=2701: Pattern 73.
The Greek value of Jesus, 888=24×37: Pattern 78.

Uncounted curiosities and experimental observations

These items are retained for transparency and interest. They do not contribute to the comparative challenge.

Observation	Why it is uncounted
In The Big Bang Theory, Sheldon Cooper calls 73 his favorite number.	Popular culture; its mathematical explanation repeats observations already listed.
37 is often perceived as a “random-looking” number in informal demonstrations.	Selection rates depend on the prompt, range, and participants. The former “one in three” claim was not confirmed.
The American Kennel Club permits 37 dogs of each breed to share a registered name. 37 is XXXVII, while 38 is XXXVIII.	A changeable organizational rule. The legacy six-character-field explanation is widely reported but not stated in the current official rule.
An illustrated stepped-block arrangement shows 16+12+9=37 visible unit squares.	A visual construction, not a complete 4×4×4 cube. A clearer independent construction diagram is still desirable.
Every factorial from 6! onward has digital root 9.	A general property applying equally to 25!, 37!, 52!, and 73!.
A repeated-777 sum-of-three-cubes search found offset 189=3³×7 optimal within the tested range.	sotc.py tests 7–106 groups of 777 and offsets 1–1000. It is a bounded experiment, not an infinite theorem.
The reversed Genesis word-place concatenation has a 300-term Collatz sequence.	299 operations, or 300 terms including the start and final 1; it does not distinguish 37/73 from the controls.
Standard-word-value digits and ordinal-letter-position digits each sum to 82=3⁴+1.	A valid Genesis curiosity that does not involve 37 or 73.
The binary numeral 111₂ equals 7, while its displayed digits sum to 3.	A disclosed theological number play, not a property of 37 or 73.
One traditional Gospel harmony lists 37 miracles of Jesus.	The count depends on how parallel accounts, group healings, and the resurrection are treated.

Experimental digit searches

The following matches are reproducible, but they involved choices among target strings, constants, roots, reversals, starting positions, intervals, or verse references. No simple probability is claimed.

Source	Disclosed extraction	Result
∛333	Fractional positions 35,47,59,71,83,95,107,119	31015211, the reversed ordinal-position string for אלהים.
∛556, from Psalm 55:6	Fractional positions 20,24,28,32,36,40,44	2701298=2701\|298.
∛37+∛73	Start at fractional position 3355; take every 777th digit	4555828, the shared word-by-word digital-root sequence.
3−√5	Zero-based fractional start 21571; interval 7527, derived by splitting 215717527	5300401040, the concatenated standard values of “the heavens.”
∛144+∛549	Complete-string zero-based index 5273, equivalent to fractional position 5271	Contiguous 2201211022, the ordinal positions of bereshit.

Claims removed during review. The replacement page omits unconfirmed claims about 37 hand muscles, number counts in the Gospels and Exodus, word frequencies, and an undefined alphabetical ordering of Roman numerals. It also removes the open-ended demand to match every factoid on an external website.

This page distinguishes exact mathematics, interpretive associations, general curiosities, exploratory searches, and material already counted elsewhere. That separation is part of the published challenge standard.