4x^2 -4xy + 7y^2 Problem

[Artur]

Differences between two positive cubes
A038593 <http://www.research.att.com/%7Enjas/sequences/A038593>
A014439 <http://www.research.att.com/%7Enjas/sequences/A014439>
A014440 <http://www.research.att.com/%7Enjas/sequences/A014440>
A014441 <http://www.research.att.com/%7Enjas/sequences/A014441>
Artur


David Wilson pisze:
> ----- Original Message ----- From: "David Harden" <oddleehr at alum.mit.edu>
> To: <seqfan at ext.jussieu.fr>
> Sent: Monday, April 28, 2008 12:43 PM
> Subject: 4x^2 -4xy + 7y^2 Problem
>
>
>> [[Complete argument omitted for brevity]]
>>
>> Since there are no residue classes mod 24 produced by more than one 
>> of those quadratic forms, the necessary congruential conditions for p 
>> to be produced by one of those forms are also sufficient. Therefore 
>> any p == 1 or 7 mod 24 has p = x^2 + 6y^2 for some integers x and y. 
>> I think this is what David Wilson wanted. In any case, the rest of 
>> the argument can proceed using elementary modular arithmetic.
>>
>> ---- David
> Given Dave Harden's argument that p == 1 or 7 (mod 24) <=> p = x^2 + 
> 6y^2 (the hard part, thank you Dave), I supplied the easy rest of the 
> argument that p == 7 (mod 24) <=> p = 4x^2-4xy+7y^2 in an earlier note.
>
> The relevant sequence can be changed to "Primes == 7 (mod 24)" or 
> "Primes of form 24k+7".
>
> On an unrelated note, interesting sequences that may not be in the 
> OEIS are "differences of cubes" and "differences of positive cubes".
>
>
>
>
>
>
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David Wilson asked (implicitly) which sequences in the OEIS 
involve the difference between two cubes.  I did a search 
and found a few - I will make an entry in the Index
under "difference of two cubes".  There are probably a lot
that I missed, though.    Neil
%N A038673 Numbers that are not the difference between two positive cubes.
%N A038593 Differences between positive cubes in 1, 2 or 3 ways: union of A014439, A014440 and A014441.
%N A014439 Differences between two positive cubes in exactly 1 way.
%N A038847 Odd numbers that are differences between two cubes in at least one way.
%N A038862 Numbers n such that n ends with '7' and is difference between two cubes in at least one way.
%N A038855 Numbers that are divisible by 7 and are differences between two cubes in at least one way.
%N A098110 Smallest number that is the difference between two positive cubes in n ways.
%N A038808 Palindromic numbers which are the difference of two positive cubes.
%N A085367 Semiprimes that can be expressed as the sum or difference of two cubes. Common terms of A001358 and A045980.
%N A038598 Differences between numbers that are a difference between 2 positive cubes in at least one way.
%N A038597 Numbers n such that n^2 is a difference between 2 positive cubes in at least one way.
%N A086121 Positive sums or differences of two cubes of primes.
%C A038632 The first number that is a difference of 2 cubes in 4 ways > 3000000
%N A038864 Numbers n such that n ends with '9' and is difference between two cubes in at least one way.
%N A038848 Even numbers that are differences between two cubes in at least one way.
%N A038861 Numbers n such that n ends with '6' and is difference between two cubes in at least one way.
%C A085377 Geometrically, 13^2 = 8^3 - 7^3 means that the square of the hypotenuse of a Pythagorean triangle (5,12,13) is the difference of two cubes, which I recently found on p70 of David Wells' book "The Penguin Dictionary of Curios and Interesting Numbers", Penguin Books, 1997. See also A085479.
%N A038849 Numbers that are divisible by 4 and are differences between two cubes in at least one way.
%N A038850 Numbers that are divisible by 8 and are differences between two cubes in at least one way.
%N A038856 Numbers n such that n ends with '1' and is difference between two cubes in at least one way.
%N A038851 Divisible by 3 (and 9) and are differences between two cubes in at least one way.
%N A038858 Numbers n such that n ends with '3' and is difference between two cubes in at least one way.
%N A129965 Triangular numbers which are differences of nonnegative cubes.
%N A038863 Numbers n such that n ends with '8' and is difference between two cubes in at least one way.
%N A038859 Numbers n such that n ends with '4' and is difference between two cubes in at least one way.
%N A038857 Numbers n such that n ends with '2' and is difference between two cubes in at least one way.
%N A038596 Numbers n such that n is a perfect square and is a difference between 2 positive cubes in at least one way.
%N A038853 Numbers that are divisible by 5 and are the difference between two (different positive) cubes in at least one way.
%N A038860 Numbers that end in '5' and are the difference between two (positive) cubes in at least one way.
%N A038594 Numbers n such that n and n+1 are differences between 2 positive cubes in at least one way.
%N A038595 Numbers n such that n and n-1 are differences between 2 positive cubes in at least one way.
%N A038852 Numbers that are divisible by 6 (and 18) and are differences between two cubes in at least one way.
%N A034179 Difference between two positive cubes in more than one way.
%N A014440 Differences between two positive cubes in exactly 2 ways.
%N A038854 Numbers that are divisible by 10 and are differences between two cubes in at least one way.
%N A014441 Differences between two positive cubes in exactly 3 ways.
%N A051393 Numbers whose square is expressible as the difference of positive cubes in more than one way.
%N A125063 Numbers expressible as sum or difference of two cubes of primes in at least two ways.