Expressing difference of squares as diophantine equation

Question

I have a function defined by $f=(2c-1)^2-2b^2$. I want to express this in the form of a linear diophantine equation. I tried $f_1=8m\pm 1$ for some $m$, but it unfortunately does not work since for example $f=65$ never occurs for any value of $b$ or $c$. So, the question is can such a linear diophantine relation be written?

Answer 1

Well, you have forced the represented numbers to be odd, indeed $\pm 1 \pmod 8 \; .$

So that is it, an odd number is represented if and only if all prime factors $q \equiv \pm 3 \pmod 8$ occur with even exponents. Indeed, if there are any such $q,$ we must have $q | b$ and $q| (2c-1).$ In your example, $65 = 5 \cdot 13,$ each with odd exponent $1,$ each $5 \pmod 8.$

In particular, any odd number, all of whose factors are $\pm 1 \pmod 8 \; ,$ can be expressed this way

Why not: here are the primes $p = u^2 - 2 v^2$ up to 1000. The prime $2$ is included here by the program, although your polynomial cannot actually represent any even number.

jagy@phobeusjunior:~$ ./Conway_Positive_Primes 1 0 -2   1000   8
           1           0          -2   original form 

           1           2          -1   Lagrange-Gauss reduced 



 Represented (positive) primes up to  1000

     2     7    17    23    31    41    47    71    73    79
    89    97   103   113   127   137   151   167   191   193
   199   223   233   239   241   257   263   271   281   311
   313   337   353   359   367   383   401   409   431   433
   439   449   457   463   479   487   503   521   569   577
   593   599   601   607   617   631   641   647   673   719
   727   743   751   761   769   809   823   839   857   863
   881   887   911   919   929   937   953   967   977   983
   991

=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=   
 these are the collection of remainders when dividing by   8

      1      2      7



 Represented (positive) primes up to  1000  and value mod    8

           1           0          -2   original form 

A good book for self study of number theory, currently on sale at about $42 I guess shipping also costs.