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# Five peculiar theorems on simultaneous representation of primes by quadratic forms

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## Abstract

It is a theorem of Kaplansky that a prime p ≡ 1 (mod 16) is representable by both or none of x² + 32y² and x² + 64y², whereas a prime p ≡ 9 (mod 16) is representable by exactly one of these binary quadratic forms. In this paper five similar theorems are proved, and a heuristic argument is given why there are no other results of the same kind. The latter argument relies on the (plausible) conjecture that the list of 485 known negative discriminants Δ such that the class group C(Δ) has exponent 4 is complete. The methods are purely classical.

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... Theorem 1 in section 2 below fills this gap. This investigation was motivated by [2] where the problem occurred of determining the Rédei field over K = Q( √ −7) for D = −1792. ...
... It also follows that the classes [f 1 ], . . . , [f µ ] generate C [2] and satisfy one non-trivial relation. ...
... The group of characters χ : C → C × is denoted C * . The elements of the subgroup C * [2] = {χ ∈ C * | χ 2 = 1} are called quadratic characters. A prime discriminant is a number of the form -4, 8, -8 or p * = (−1) (p−1)/2 p where p is any odd prime. ...
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For an arbitrary non-square discriminant D, the Rédei field Γ0(D) is introduced as an extension of K = Q(√D) analogous to the genus field and connected with the Rédei-Reichardt Theorem. It is shown how to compute Rédei fields, and this is used to find socles of dyadic extensions of K for negative D. Finally, a theorem and two conjectures are presented relating the fields Q(√(–p)) and Q(√(–2p)) for an odd prime p.
... We found some of the polynomials included here by computational experimentations. For more details in this direction see [1], [2], [5], [6], [7], [15] and [19]. has a solution (XXX) (q = 30) (XXXI) (q = 31) (L. ...
... For further developments similar to Kaplansky's result we refer to [2]. One can show that the representations in Theorem 1.1 are unique (see Problem 3.23 in [6]). ...
... [6], Lemma 1.4, p. 10) here. If we calculate M = (2b)2 [3 2 + 17(1) 2 ] − 4q = [3(2b) − 2a][3(2b) + 2a], we see that 2(13) divides M and so it divides either 3(2b) − 2a or 3(2b) + 2a. Without loss of generality we may assume that 2(13) divides 3(2b) − 2a. ...
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Representations of primes by simple quadratic forms, such as ±a2±qb2, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gauss and many others. We are interested in a comprehensive list of such results, for q � 20. Some of the results can be established with elementary methods and we exemplify that in some instances. We are introducing new relationships between various representations.
... Theorem 1 in section 2 below fills this gap. This investigation was motivated by [2] where the problem occurred of determining the Rédei ...
... A form (a, b, c) is called ambic (or ambiguous) if a divides b, and a class of forms is called ambic if it contains an ambic form. As shown by Gauss, the set of ambic classes coincides with the subgroup C [2] = {a ∈ C | a 2 = 1} of self-inverse classes, and each ambic class contains exactly two ambic forms (a, b, c) with b equalling 0 or a. We adopt the terminology of [10] and call such forms ancipital.Table 1 defines, depending on D's congruence class, a number µ and a list of ancipital forms f 1 , . . . ...
... It also follows that the classes [f 1 ], . . . , [f µ ] generate C [2] and satisfy one non-trivial relation. The group of characters χ : C → C × is denoted C * . ...
Article
For an arbitrary non-square discriminant D, the R edei eld 0(D) is introduced as an extension of K = Q( p D) analogous to the genus eld and con- nected with the R edei-Reichardt Theorem. It is shown how to compute R edei elds,
... Kaplansky proved his theorem using two well-known results: 2 is a 4th power modulo a prime p if and only if p is represented by x 2 + 64y 2 (Gauss [7, p. 530]) and −4 is an 8th power modulo a prime p if and only if p is represented by x 2 + 32y 2 (Barrucand and Cohn [3]). Using class field theory, Brink [4] was able to prove five more theorems similar to that of Kaplansky In [4], Brink claims that these are the only results of their kind and gives a heuristic argument as support. As an example, Brink shows that there is no similar result for primes represented by x 2 + 128y 2 and x 2 + 256y 2 . ...
... Kaplansky proved his theorem using two well-known results: 2 is a 4th power modulo a prime p if and only if p is represented by x 2 + 64y 2 (Gauss [7, p. 530]) and −4 is an 8th power modulo a prime p if and only if p is represented by x 2 + 32y 2 (Barrucand and Cohn [3]). Using class field theory, Brink [4] was able to prove five more theorems similar to that of Kaplansky In [4], Brink claims that these are the only results of their kind and gives a heuristic argument as support. As an example, Brink shows that there is no similar result for primes represented by x 2 + 128y 2 and x 2 + 256y 2 . ...
Article
Kaplansky [2003] proved a theorem on the simultaneous representation of a prime $p$ by two different principal binary quadratic forms. Later, Brink found five more like theorems and claimed that there were no others. By putting Kaplansky-like theorems into the context of threefield identities after Andrews, Dyson, and Hickerson, we find that there are at least two similar results not on Brink's list. We also show how such theorems are related to results of Muskat on binary quadratic forms
... are equivalent. Combining this with the above gives that also Five results similar to Kaplansky's theorem were found in [3], for example the following: A prime p ≡ 1 (mod 20) is representable by both or none of x 2 + 20y 2 and x 2 + 100y 2 , whereas a prime p ≡ 9 (mod 20) is representable by exactly one of these forms. The proof used class field theory. ...
Article
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We give a new proof of a recent theorem of Kaplansky and use it to revive an old, seemingly forgotten result of Glaisher.
... Hence if a 4 + 2a 2 + 2 ≡ 0 (mod p) then (ab) 4 − 2(ab) 2 + 2 ≡ 0 (mod p) which shows that the equation x 4 − 2x 2 + 2 ≡ 0 (mod p ) always has a solution. Also, another classical result along these lines is Kaplansky's Theorem ([14]): For further developments similar to Kaplansky's result we refer to [2]. One can show that the representations in Theorem 1.1 are unique (see Problem 3.23 in [6]). ...
Article
Full-text available
Representations of primes by simple quadratic forms, such as $\pm a^2\pm qb^2$, is a subject that goes back to Fermat, Lagrange, Legendre, Euler, Gauss and many others. We are interested in a comprehensive list of such results, for $q\le 20$. Some of the results can be established with elementary methods and we put them at work on some instances. We are introducing new relationships between various representations.
Article
We study intrinsic Galois structure behind theorems by Kaplansky, Brink, and Mortenson on simultaneous representation of primes by binary quadratic forms with different discriminants. This study brings new theorems like theirs also in indefinite quadratic forms.
Article
Two basic approaches have been used to develop explicit formulae for the number of classes in a genus of binary quadratic lattices over an algebraic number field. Analytic machinery in the form of the Minkowski-Siegel Mass Formula or the Tamagawa number of an algebraic group was employed by Pfeuffer [13] and Shyr [17] to obtain such a formula for maximal positive definite lattices over totally real number fields. On the other hand, Peters [10] observed that a formula applicable to maximal lattices over any number field can be deduced by algebraic methods from the theory of quadratic field extensions. Using group-theoretic techniques set up by the present authors [3] along with the calculation of certain local unit indices, Korner [6] derived the corresponding formula for non-maximal lattices.(Received March 17 1981)
The forms x + 32y 2 and x + 64y 2 [sic
• I Kaplansky
I. Kaplansky, The forms x + 32y 2 and x + 64y 2 [sic], Proc. Amer. Math. Soc. 131 (2003), no. 7, 2299-2300.
The Groups of Order 2 n (n ≦ 6), The MacMillan Company
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The forms x + 32 y 2 and x + 64 y 2 [sic
• I Kaplansky
I. Kaplansky, The forms x + 32 y 2 and x + 64 y 2 [sic], Proc. Amer. Math. Soc. 131 (7) (2003) 2299-2300.
The groups of order 2 n
• M Hall
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M. Hall, Jr. and J. K. Senior, The groups of order 2 n (n 6), The MacMillan Company, New York, 1964.