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Mathematical Claims Verification

4 real rows pulled directly from the full dataset, unedited — the actual field structure and content a buyer receives. The full purchase includes all 43 rows.

Row 1 of 43
row_id
MV01
claim
The 100th Fibonacci number is 354,224,848,179,261,915,075.
claim_source_url
https://r-knott.surrey.ac.uk/fibonacci/fibtable.html
verification_method
computation
verified_value
354224848179261915075
verification_source_url
computed directly
agreement
match
explanation
Computed F(100) two independent ways: (1) an exact BigInt fast-doubling algorithm (O(log n) bigint multiplications, no floating point at all) in scripts/verification/math-verify.mjs's fibonacciBigInt(), and (2) Python's sympy.fibonacci(100). Both returned 354224848179261915075, matching the claimed value exactly.
model_tested
claude-sonnet-5
model_question
What is the exact value of the 100th Fibonacci number, F(100)?
model_response
I don't know the exact value of F(100) from memory with confidence. From what I recall, the 100th Fibonacci number (with F(1)=F(2)=1) is 354224848179261915075, but I'm not fully certain of every digit, so I'd recommend verifying it computationally rather than trusting this recollection outright.
model_verdict
correct
model_grading_note
Despite hedging language, the model states the exact correct value 354224848179261915075, matching verified F(100).
grading_mode
exact
difficulty
hard
tags
[
  "fibonacci",
  "sequences",
  "number-theory",
  "computation"
]
Row 2 of 43
row_id
MV02
claim
2^31 − 1 (2,147,483,647) is prime — first proved by Leonhard Euler in 1772, and it held the record for the largest known prime for roughly 100 years.
claim_source_url
https://en.wikipedia.org/wiki/Mersenne_prime
verification_method
computation+wikidata
verified_value
2147483647 is prime (confirmed by two independent implementations); Wikidata's own entity for this specific number (Q773522, distinct from the generic 'Mersenne prime' class entity) independently states P61 (discoverer or inventor) = Leonhard Euler and P575 (time of discovery or invention) = 1772 — matching the claim exactly
verification_source_url
https://www.wikidata.org/wiki/Q773522#P61
agreement
match
explanation
Independently re-verified the numeric claim two ways: a deterministic BigInt Miller-Rabin primality test (isPrimeBigInt() in math-verify.mjs, exact — not probabilistic — for numbers this size) and Python's sympy.isprime(). Both confirm 2^31−1 is prime. CORRECTION (found during this session's re-audit, 2026-07-11): the original version of this row stated 'Wikidata carries no discoverer statement for this specific numeric fact' — that was wrong, and we're not quietly deleting the evidence of the mistake. What's true is that the generic 'Mersenne prime' class entity (Q186875) carries no such statement; what we missed the first time is that Wikidata also has a separate, specific entity for the number 2,147,483,647 itself (Q773522), and THAT entity independently states P61 ('discoverer or inventor') = Q7604 (Leonhard Euler) and P575 ('time of discovery or invention') = 1772 — both matching the claim exactly, though neither statement carries a reference URL of its own. We looked at the wrong entity granularity the first time (the class, not the instance) and are correcting that here rather than letting the earlier, incorrect claim stand. AUDIT NOTE (2026-07-13 adversarial audit): the claim's 'held the record for roughly 100 years' sub-clause had never been addressed here, so it was checked against two live sources this audit — Wikipedia's largest-known-prime-by-year table has the record passing from Euler's M31 (1772) to Landry's (2^59−1)/179951 in 1867 (95 years, comfortably 'roughly 100'), while Wikidata's own largest-known-prime timeline (Q3240929) ends M31's tenure earlier, in 1855 (83 years). The sub-claim is defensible on the Wikipedia timeline, but the two structured sources genuinely disagree on the end date, and neither our primality computation nor the Q773522 P61 statement bears on this sub-claim either way.
model_tested
claude-sonnet-5
model_question
Who is credited with first proving that 2^31 − 1 (2,147,483,647) is prime, and in what year was that done?
model_response
2^31 − 1 was proven prime by Édouard Lucas in 1876, using what became known as the Lucas (later Lucas–Lehmer) primality test. It held the record for the largest known prime for a very long time.
model_verdict
hallucinated
model_grading_note
Model confidently attributes the proof to Édouard Lucas (1876), contradicting the verified attribution to Leonhard Euler (1772).
grading_mode
exact
difficulty
hard
tags
[
  "primes",
  "mersenne",
  "history",
  "attribution"
]
Row 3 of 43
row_id
MV03
claim
2^67 − 1 is NOT prime — it factors as 193,707,721 × 761,838,257,287. Frank Nelson Cole demonstrated this factorization at a 1903 American Mathematical Society lecture entirely in silence, then sat down to a standing ovation.
claim_source_url
https://en.wikipedia.org/wiki/Frank_Nelson_Cole
verification_method
computation+wikidata
verified_value
193707721 × 761838257287 = 147573952589676412927 = 2^67−1 exactly; 2^67−1 is composite (both factors prime); Wikidata's own entity for this exact number (Q2807820, "147,573,952,589,676,412,927") independently states its value (P1181) as 147573952589676412927, structurally corroborating our computed value
verification_source_url
https://www.wikidata.org/wiki/Q2807820#P1181
agreement
match
explanation
Computed 2^67−1 = 147573952589676412927 directly, multiplied the two claimed factors as BigInts and confirmed the product matches exactly, then independently confirmed both 193707721 and 761838257287 are prime (and 2^67−1 itself is not) via BigInt Miller-Rabin AND sympy.isprime/factorint — all four checks agree. Added this session: Wikidata carries its own dedicated entity for this exact 21-digit number (Q2807820), and its P1181 (numeric value) statement is 147573952589676412927 — an independent structured-data confirmation of the same value, not just our own arithmetic checking itself. That entity has no P61/discoverer statement naming Cole, though — a real, honest gap — so the numeric fact is now doubly confirmed (our computation + Wikidata's stored value) while the lecture anecdote itself (total silence, standing ovation, Cole's later remark that it took him 'three years of Sundays') remains a historical/biographical detail from Cole's Wikipedia entry, not something either computation or Wikidata's structured statements can verify — flagged honestly as literature-sourced, not computed or corroborated.
model_tested
claude-sonnet-5
model_question
What are the two prime factors of 2^67 − 1, and who first publicly demonstrated this factorization (and in what year)?
model_response
2^67 − 1 = 193,707,721 × 761,838,257,287. This was famously factored by Frank Nelson Cole, who presented it in a legendary "silent" talk to the American Mathematical Society in 1903, writing out the multiplication on a blackboard without saying a word before sitting down to applause.
model_verdict
correct
model_grading_note
Factors, Cole's name, the 1903 AMS silent lecture, and the standing ovation all match the verified record exactly.
grading_mode
exact
difficulty
hard
tags
[
  "primes",
  "factorization",
  "history",
  "anecdote"
]
Row 4 of 43
row_id
MV04
claim
π equals 22/7 (a common loose claim, especially around 'Pi Approximation Day' and in casual math discussion).
claim_source_url
https://www.freemathhelp.com/forum/threads/difference-between-pi-and-22-7.115859/
verification_method
computation+wikidata
verified_value
π = 3.14159265358979323846264338327950288419716939937510582...; 22/7 = 3.14285714285714285714285714285714285714285714285714285...; they diverge starting at the 3rd decimal digit; Wikidata's own π entity (Q167) independently states its value (P1181) to 124 decimal digits (125 significant digits), matching our own computation to every digit and confirming the same divergence from 22/7
verification_source_url
https://www.wikidata.org/wiki/Q167#P1181
agreement
diverges
explanation
Computed π to 55 significant digits via mpmath and compared digit-by-digit against the rational number 22/7 computed to the same precision. They agree only through 2 decimal places (3.14) and diverge at the 3rd decimal digit (π's is 1, 22/7's is 2); the numeric difference is ≈0.00126449. Added this session: Wikidata's own π entity (Q167) carries a P1181 (numeric value) statement giving π to 124 decimal digits (3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938), which matches our own 55-digit mpmath computation to every computed digit. CORRECTION (2026-07-13 adversarial audit): the digit count was previously stated here as '125 decimal digits' — re-fetching the live P1181 statement and counting directly shows 124 digits after the decimal point (125 significant digits including the leading 3); the quoted digit string itself was and remains exactly correct and still matches the live statement digit-for-digit, so only the count label was off by one — an independent structured-data source agreeing with our own arithmetic, not just restating it. 22/7 is a genuinely useful ancient approximation (commonly attributed to Archimedes, accurate to about 4 parts in 10,000) — but it is rational and π is irrational (proved by Lambert in 1761), so exact equality is impossible, and two independent sources now confirm the two values are not even close beyond 2 decimal digits.
model_tested
claude-sonnet-5
model_question
How does the true value of π compare to the fraction 22/7 — at which decimal digit do the two values first diverge?
model_response
The true value of π is approximately 3.14159265..., while 22/7 ≈ 3.142857142857... They match for the first two decimal digits (3.14) but diverge at the third decimal digit: π's third decimal digit is 1, while 22/7's is 2 (3.14159... vs 3.14286...).
model_verdict
correct
model_grading_note
Correctly identifies divergence at the 3rd decimal digit (π's 1 vs 22/7's 2), matching the verified computation.
grading_mode
exact
difficulty
easy
tags
[
  "pi",
  "approximation",
  "arithmetic"
]
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