Verified calculations involving 37, 73, 23, and mathematical constants
This page records 32 distinct observations retained from the original
44-item prime-pair collection. The calculations were reproduced using the
digit strings and conventions used by mathematics.htm. Duplicate
restatements and material already counted in the first 100 patterns were
consolidated or moved to the cross-reference section.
The decimal point is removed. The initial integer digit is position 1, so π begins 314159…, e begins 271828…, and φ begins 161803…. “Between positions a and b” excludes both endpoints; “positions a through b” includes both endpoints.
The first 23 digits of π form a complete integer divisible by 73. Their digit sum is 111 = 3 × 37 = 777 ÷ 7.
Verify divisibility by 73 · Verify the digit sum. The separate fact that 23 Genesis word-value subsets are divisible by 37 belongs to Pattern 40.
The integer formed from the first 23,023 digits of π is divisible by 23.
The integer formed from the first 73,073 digits of π is divisible by 73.
The complete integer is divisible by 73, and its digit sum is 33,005, which is divisible by 23.
Because 259 = 37 × 7, this range uses positions 2 through 258. Its digit sum is 1,168, which is divisible by 73.
The integer formed from positions 2 through 72 of π is divisible by 23.
The digit sum of the first 7! = 5,040 digits is 22,866, which is divisible by 37.
The integer formed from positions 2 through 3,772 of π is divisible by 23.
Because 37 = 2,187, the range contains positions 2 through 2,186. Its digit sum is exactly 10,001 = 73 × 137.
The complete integer formed from positions 2 through 7,824 is divisible by 37. The endpoint 7,825 is a separately selected number and is disclosed as such.
For endpoints 9, 16, and 78: the integer at positions 2–8 is divisible by 23; the digit sum at positions 2–15 is 74 and divisible by 37; and the first 78 digits have sum 368 and are divisible by 23. These three disclosed selections form one compound observation.
The digit sum for positions 2 through 32,992 is 148,603, which is divisible by 23. The endpoint 32,993 is a separately selected number.
Their digit sum is exactly 1,679 = 23 × 73. Divisibility by 23 and 73 is one fact about the same sum and is counted once.
Their digit sum is 22,200, divisible by both 37 and 888. The complete 4,900-digit integer is divisible by 73. These properties of the same selected range form one compound observation.
The integer formed from positions 2 through 2,700 leaves remainder 23 when divided by 37.
The inclusive 37-digit integer is divisible by 37. The same digits have sum 207, which is divisible by 23. These are counted as one selected-range observation.
The exact SI value 299,792,458 metres per second, with its digits treated as an integer, is divisible by 73. This is unit- and definition-dependent and is not a dimensionless universal-number result.
299,792,458 ÷ 2,701 = 110,993 with remainder 365. The quotient's digits sum to 23. The appearances of 365 and 23 are interpretive associations, and the SI-unit dependence remains disclosed.
Verify quotient and remainder · Verify the quotient's digit sum.
The digit sum for positions 2 through 776 is 3,473. It is divisible by 23 and ends in 73.
The integer formed from positions 2 through 776 of e is divisible by 37.
The digits of 299,792,458 sum to 55. The π digits strictly between positions 1 and 55 have sum 259 = 7 × 37.
The inclusive fifteen-digit range has digit sum exactly 73.
Because 851 = 23 × 37, this endpoint combines the chromosome-pair number with 37. The digit sum is 3,795, divisible by 23.
The remainder is 27,163,154, which is divisible by 73. This observation depends on the selected SI integer and is therefore unit-dependent.
√(7.3π) ≈ 22.7041, which rounds to 23. It does not equal 23 exactly.
Positions 37 through 43 form 4,197,169. Its digits sum to 37, and 4,197,169 = 37 × 113,437; the cofactor also ends in 37. The earlier probability and control-impossibility claims are not included because they were not established by the calculator.
The first five digits of π, 31,415, added to 299,792,458 produce 299,823,873, which ends in 73. This is unit-dependent.
Dividing the 23-digit π integer by 27,182 leaves remainder 17,770, whose middle three digits are 777. The description is a digit-pattern observation rather than an additional divisibility result.
The endpoint 23,203 is formed by appending the alphabet-position value 23 to the traditional value 203 of the second Genesis 1:1 word. The 23,203-digit π integer leaves remainder 333 when divided by 777.
The first seven digits of π leave remainder 37 when divided by 73.
The first seven digits of the golden ratio, φ, leave remainder 23 when divided by 37.
The first 37 digits of π leave remainder 37 when divided by 73.
| Original entry | Disposition |
|---|---|
| 2 and 3 | Already combined and counted as Pattern 44: ⌈πe⌉ and ⌊eπ⌋ equal 23. |
| 9 | Already counted as Pattern 69. |
| 18 | Removed. The unexplained sequence 6, 6, 120, and 30,240 could not be reconstructed from the stated perfect-number rule. |
| 19 | Already counted as Pattern 70. |
| 24 | Uses the same verse-position/value/position calculation as Pattern 32. Its additional divisibility by 73 may be noted there but is not counted again here. |
| 25 | Already counted through Patterns 33 and 72, and documented in the revised Bonus collection. |
| 26 | A bundle of Genesis 1:1 calculations already represented among the first 100 patterns. |
| 36 | Removed as outside this category. Also, 2997 ÷ 954 = 3.141509…, not the same five-significant-digit π result used elsewhere. |
The original comparison data are retained below. They list endpoints under 1,000 that satisfy the selected divisibility condition. This exploratory table demonstrates that the control numbers 25 and 52 also have many numerical matches; it is not a probability calculation and does not make every listed endpoint a separate pattern.
Loading comparison values…“23 chromosomes” on the original page has been replaced where appropriate by “23 chromosome pairs in a typical human cell.” Speed-of-light observations use the exact SI integer in metres per second. They depend on that unit definition and are labeled accordingly.