An Unsolvable Problem of Elementary Number Theory
American Journal of Mathematics, 1936 路 paper 路 Mathematics and statistics
Proposes a precise definition of effective calculability, corresponding to the intuitive notion used in number-theoretic problems. Shows by an example that not every problem of this kind, which asks for an effectively calculable function, has a solution.
Checked: PDF from cis.upenn.edu, fetched 4 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance Last display on p. 359 prints "{位x 路 饾敪(位n 路 饾敜(饾敚(x, n), 1, 1))}(c)", with a misplaced closing parenthesis: 饾敜 receives three arguments and 饾敪 only one; the evident intended form is {位x 路 饾敪(位n 路 饾敜(饾敚(x, n)), 1, 1)}(c).
- p. 16 Last displayed formula on the page (proof of Lemma)
{位x 路 饾敪(位n 路 饾敜(饾敚(x, n), 1, 1))}(c)
- p. 17 Displayed formula for 饾敗(2), proof of Theorem XVIII
饾敗(2) conv 位u 路 饾敔(饾敚(u, 饾敪(位m 路 饾敜(饾敚(u, m)), 1))).
- p. 16 Earlier displayed formula in the same proof
{位xy 路 饾敪(位n 路 未(饾敚(x, Z1(n)), 饾敚(y, Z2(n))), 1)}(a, b)
Judge's note: Viewed PDF pages 16 and 17. The page image shows the grouping 饾敜(饾敚(x, n), 1, 1)) with two closing parentheses at the end, so this is not an extraction artefact. G is defined just above as a function of one positive integer, and the other two displays give 饾敪 a function and a starting value 1: 饾敪(位n 路 未(...), 1) on p. 359 and 饾敪(位m 路 饾敜(饾敚(u, m)), 1) on p. 360. As printed, the inner term 饾敜(饾敚(x, n), 1, 1) reduces to 1 whatever G returns (G is total, value 1 or 2), so the formula could not distinguish whether C has a normal form. With the parenthesis moved, 饾敪(..., 1) yields a numeral exactly when C has a normal form, and that numeral applied to 1 converts to 1, matching the next sentence ("convertible into the formula 1 if and only if C has a normal form"). Rated S1 because the displayed formula is wrong as printed, though the fix is evident.
Second check: another model agreed
Report a problem with this finding
Reference or label Proof of the Lemma (p. 359) twice cites Theorem X ("For, by Theorem X, the formula a (the formula b), which stands for the positive integer which is the G枚del representation of the formula A ... can be expressed as a recursive function of the formula A"; "Again, by Theorem X, the formula c ..."), but the map from a positive integer to the corresponding formula 1, 2, 3, ... is Theorem IX.
- p. 16 Proof of Lemma, first paragraph
For, by Theorem X, the formula a (the formula b), which stands for the
- p. 16 Proof of Lemma, third paragraph
Again, by Theorem X, the formula c, which stands for the positive
- p. 11 Theorem IX
The function, whose value for each of the positive integers 1, 2, 3, 路 路 路 is the corresponding formula 1, 2, 3, 路 路 路 , is recursive.
- p. 17 Proof of Theorem XVIII, paragraph on Z^-1
By Theorem IX there exists a recursive function Z-1 of one positive
Only the authors can say what is right here.
Judge's note: Viewed PDF pages 11, 12, 16 and 17. Theorem IX (p. 354) gives positive integer -> corresponding formula 1, 2, 3, .... Theorem X (pp. 354-355) gives only the reverse direction, formula 1, 2, 3, ... -> positive integer (or integer plus one). The Lemma's step needs the numeral formula a as a function of A, which under the paper's G枚del-representation convention is integer -> formula, i.e. Theorem IX. The paper uses the theorems in exactly these senses on p. 360: Theorem IX for Z^-1 (integer -> G枚del representation of the corresponding formula) and Theorem X for C (formula -> integer plus one). Both citations on p. 359 clearly print "Theorem X". Bucketed as an author query rather than a correction because it rests on a reading of mathematical intent; an indirect use of X is hard to construct but only the author could rule it out.
Second check: another model agreed
Report a problem with this finding
4 minor (typos and the like; most checked by the judge only)
Typo Footnote 18 prints "the relationship betwen effective calculability and recursiveness"; "betwen" should be "between".
- p. 13 Footnote 18
relationship betwen effective calculability and recursiveness
Judge's note: Viewed the crop of footnote 18 on PDF page 13. The print reads "betwen" in the first line, while "between" is spelled correctly in the fourth line of the same footnote. Not an extraction artefact.
Second check: none (judge only)
Report a problem with this finding
Typo Missing relative pronoun on p. 18: 'convertible into the formula stands for the G枚del representation of a formula definitely not convertible into e(n)' should read 'the formula which stands for'.
- p. 18 Proof of Theorem XVIII, paragraph beginning 'By Theorem III'
convertible into the formula stands for the G枚del representation of a formula definitely not
- p. 18 Proof of Theorem XVIII, paragraph beginning 'By Theorem III'
formula stands for the G枚del representation of a formula definitely not con-
- p. 18 paragraph after the rules (1)-(3)
convertible into the formula stands for the Godel representation
- p. 18
convertible ilto the formula stands for the Godel representation
Judge's note: I checked the crop and the page image of p. 18. The line ends 'convertible into the' and the next line begins 'formula stands for the G枚del representation', so the word is missing from the print itself and was not dropped by extraction. Two lines earlier the parallel clause reads 'convertible into the formula which stands for the G枚del representation of e(n)'. Reviewer note 4 also reports 'ilto'; that is an extraction artefact, because the image clearly reads 'into'.
Second check: none (judge only)
Report a problem with this finding
Typo Misspelling on p. 19, in the argument after Corollary 1: 'were aslo recursively enumerable' should read 'also'.
- p. 19 Proof outline after Corollary 1
which do not have a normal form were aslo recursively enumerable
- p. 19 Corollary 1 proof, 2nd paragraph
were aslo recursively enumerable
- p. 19
were aslo recursively enumerable, it would
Judge's note: I checked crop C005-a, taken from p. 19. The printed text reads 'which do not have a normal form were aslo recursively enumerable, it would', so the misspelling is in the print and is not an OCR error.
Second check: none (judge only)
Report a problem with this finding
Typo Page 363 prints "then the value of each of those terms 1)"; the verb is missing (evidently "is 1").
- p. 20 First paragraph, parenthetical
then the value of each of those terms 1)
- p. 20 First paragraph, parenthesis
of those terms 1)
- p. 20 first paragraph
the value of each of those terms 1)
Judge's note: Viewed the crop of PDF page 20. The print reads "of those terms 1)." with no verb or relation sign between "terms" and "1". F takes the values 2 or 1, and a term whose value provably cannot be calculated must have value 1, so "is 1" is the evident reading.
Second check: none (judge only)