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I'm making this an answer to make it more visible, a suggestion of Pete L. Clark that seems correct to me.

Wadim Zudilin has been running a computer program of mine on a fast computer. Today we found a string of length 11. The form is $3 x^2 + x y + 26 y^2$ of discriminant $\Delta = -311.$ The numbers represented run from 897105813710 to 897105813720. Note that this is the longest possible string for this discriminant, as the first quadratic nonresidue $\pmod {311}$ is $11.$ The first number in the string is $\equiv 0 \pmod {311},$ indeed 897105813710 = 2 * 5 * 311 * 288458461. So at this point I conjecture that there is NO general upper bound on the number of consecutive integers that can be represented by a positive form. The discriminants I have in mind are $\Delta = -p,$ where $p \equiv 7 \pmod 8$ is a prime with a large minimal nonresidue. Such primes can be found in particular among http://www.research.att.com/~njas/sequences/A000229 although not all of these are $\equiv 7 \pmod 8.$ The conjecture, to be more specific, is that for any of these desirable discriminants, there is a represented set of consecutive integers of length $N,$ where $N$ is the smallest quadratic nonresidue $\pmod p.$

Now, I admit I we do not have any sequence of length 12 or 13 or 14. But, as with Jodie Foster in "Contact," I am the scientific type who comes around to depending on faith by the end of the movie. Meanwhile the religious guy, Matthew McConaughey, comes around to accepting the scientific conclusions.

As Wadim comments, for lengths 12 and 13 we are looking at discriminant $-479,$ the first nonresidue for $479$ is 13. For lengths 14,15,16,17 we must move to $-1559.$ But it is truly astonishing how much higher we must run the target numbers as the length increases, and as the class number and absolute value of the discriminant increase. Wadim has machines available that have built-in integers up to about $10^{18}$ and that has been critical. I have a very different program design that relies on factoring, suitable for Mathematica or gp-Pari, this program being asymptotically faster. But gp-Pari is new to me.

My recent question on the Green-Tao theorem and positive quadratic forms was an attempt to get people thinking about how to prove the non-existence of a general bound for this problem.

Finally, many thanks to Wadim Zudilin.

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I'm making this an answer to make it more visible. Wadim Zudilin has been running a computer program of mine on a fast computer. Today we found a string of length 11. The form is $3 x^2 + x y + 26 y^2$ of discriminant $\Delta = -311.$ The numbers represented run from 897105813710 to 897105813720. Note that this is the longest possible string for this discriminant, as the first quadratic nonresidue $\pmod {311}$ is $11.$ The first number in the string is $\equiv 0 \pmod {311},$ indeed 897105813710 = 2 * 5 * 311 * 288458461. So at this point I conjecture that there is NO general upper bound on the number of consecutive integers that can be represented by a positive form. The discriminants I have in mind are $\Delta = -p,$ where $p \equiv 7 \pmod 8$ is a prime with a large minimal nonresidue. Such primes can be found in particular among http://www.research.att.com/~njas/sequences/A000229 although not all of these are $\equiv 7 \pmod 8.$ The conjecture, to be more specific, is that for any of these desirable discriminants, there is a represented set of consecutive integers of length $N,$ where $N$ is the smallest quadratic nonresidue $\pmod p.$

Now, I admit I do not have any sequence of length 12 or 13 or 14. But, as with Jodie Foster in "Contact," I am the scientific type who comes around to depending on faith by the end of the movie.