Return to Answer

I recommend Gauss's third proof with modifications by Eisenstein. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The proof makes no use of any mathematical discipline other than elementary number theory. That is, it uses no abstract algebra or combinatorics. Its charm is that it uses the greatest integer function in an advanced way in place of any advanced theory. The proof uses the greatest integer function to express a variety of number-theoretic concepts and statements involving integers mathematically. Then, using properties of the greatest integer function, the proof works with these concepts and statements concretely (by manipulating them mathematically) to achieve the result. Gauss invented the greatest integer function for his third proof of QR in 1808 for this very purpose. Since it was long and messy, it was simplified by Eisenstein in 1844.

How a proof is presented by the author is also important. In my online text, "A Mathematical Analysis of the Greatest Integer Function", I include a full treatment of QR. For Gauss's third proof, I used Dence. His book, "Elements of the Theory of Numbers" is an advanced text and I felt that his presentation of the proof contained gaps. While his presentation might be appropriate for his objectives and target audience, I felt that it would seem more like an outline to undergraduate students. For this reason, when rewriting the proof, I took three measures to provide clarity. I put a lot of effort into closing every gap and providing a thorough explanation for certain parts, which accounts for a lot of its length. Since the proof has many components even after modifications by Eisenstein, I organized it by dividing it into sub-objectives and providing sub-proofs for each. I also include all three lattice-point diagrams with a numerical example in full detail. So, if you want to savor Gauss's third proof with modifications by Eisenstein, then you want to read my online text at www.greatestintegerfunctionresearch.org (Wayback Machine). My online text is also an excellent resource for various topics in number theory and analysis and is chockfull of original research.

I recommend Gauss's third proof with modifications by Eisenstein. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The proof makes no use of any mathematical discipline other than elementary number theory. That is, it uses no abstract algebra or combinatorics. Its charm is that it uses the greatest integer function in an advanced way in place of any advanced theory. The proof uses the greatest integer function to express a variety of number-theoretic concepts and statements involving integers mathematically. Then, using properties of the greatest integer function, the proof works with these concepts and statements concretely (by manipulating them mathematically) to achieve the result. Gauss invented the greatest integer function for his third proof of QR in 1808 for this very purpose. Since it was long and messy, it was simplified by Eisenstein in 1844.

How a proof is presented by the author is also important. In my online text, A Mathematical Analysis of the Greatest Integer Function, I include a full treatment of QR. For Gauss's third proof, I used Dence; his book, Elements of the Theory of Numbers is an advanced text and I felt that his presentation of the proof contained gaps. While his presentation might be appropriate for his objectives and target audience, I felt that it would seem more like an outline to undergraduate students. For this reason, when rewriting the proof, I took three measures to provide clarity. I put a lot of effort into closing every gap and providing a thorough explanation for certain parts, which accounts for a lot of its length. Since the proof has many components even after modifications by Eisenstein, I organized it by dividing it into sub-objectives and providing sub-proofs for each. I also include all three lattice-point diagrams with a numerical example in full detail. So, if you want to savor Gauss's third proof with modifications by Eisenstein, then you want to read my online text at https://euclid.uagro.mx/Greatest-INTEGER.pdf. My online text is also an excellent resource for various topics in number theory and analysis and is chock-full of original research.

I recommend Gauss's third proof with modifications by Eisenstein. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The proof makes no use of any mathematical discipline other than elementary number theory. That is, it uses no abstract algebra or combinatorics. Its charm is that it uses the greatest integer function in an advanced way in place of any advanced theory. The proof uses the greatest integer function to express a variety of number-theoretic concepts and statements involving integers mathematically. Then, using properties of the greatest integer function, the proof works with these concepts and statements concretely (by manipulating them mathematically) to achieve the result. Gauss invented the greatest integer function for his third proof of QR in 1808 for this very purpose. Since it was long and messy, it was simplified by Eisenstein in 1844.

How a proof is presented by the author is also important. In my online text, "A Mathematical Analysis of the Greatest Integer Function", I include a full treatment of QR. For Gauss's third proof, I used Dence. His book, "Elements of the Theory of Numbers" is an advanced text and I felt that his presentation of the proof contained gaps. While his presentation might be appropriate for his objectives and target audience, I felt that it would seem more like an outline to undergraduate students. For this reason, when rewriting the proof, I took three measures to provide clarity. I put a lot of effort into closing every gap and providing a thorough explanation for certain parts, which accounts for a lot of its length. Since the proof has many components even after modifications by Eisenstein, I organized it by dividing it into sub-objectives and providing sub-proofs for each. I also include all three lattice-point diagrams with a numerical example in full detail. So, if you want to savor Gauss's third proof with modifications by Eisenstein, then you want to read my online text at www.greatestintegerfunctionresearch.org (Wayback Machine. My online text is also an excellent resource for various topics in number theory and analysis and is chockfull of original research.

I recommend Gauss's third proof with modifications by Eisenstein. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The proof makes no use of any mathematical discipline other than elementary number theory. That is, it uses no abstract algebra or combinatorics. Its charm is that it uses the greatest integer function in an advanced way in place of any advanced theory. The proof uses the greatest integer function to express a variety of number-theoretic concepts and statements involving integers mathematically. Then, using properties of the greatest integer function, the proof works with these concepts and statements concretely (by manipulating them mathematically) to achieve the result. Gauss invented the greatest integer function for his third proof of QR in 1808 for this very purpose. Since it was long and messy, it was simplified by Eisenstein in 1844.

How a proof is presented by the author is also important. In my online text, "A Mathematical Analysis of the Greatest Integer Function", I include a full treatment of QR. For Gauss's third proof, I used Dence. His book, "Elements of the Theory of Numbers" is an advanced text and I felt that his presentation of the proof contained gaps. While his presentation might be appropriate for his objectives and target audience, I felt that it would seem more like an outline to undergraduate students. For this reason, when rewriting the proof, I took three measures to provide clarity. I put a lot of effort into closing every gap and providing a thorough explanation for certain parts, which accounts for a lot of its length. Since the proof has many components even after modifications by Eisenstein, I organized it by dividing it into sub-objectives and providing sub-proofs for each. I also include all three lattice-point diagrams with a numerical example in full detail. So, if you want to savor Gauss's third proof with modifications by Eisenstein, then you want to read my online text at www.greatestintegerfunctionresearch.org (Wayback Machine). My online text is also an excellent resource for various topics in number theory and analysis and is chockfull of original research.

I recommend Gauss's third proof with modifications by Eisenstein. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The proof makes no use of any mathematical discipline other than elementary number theory. That is, it uses no abstract algebra or combinatorics. Its charm is that it uses the greatest integer function in an advanced way in place of any advanced theory. The proof uses the greatest integer function to express a variety of number-theoretic concepts and statements involving integers mathematically. Then, using properties of the greatest integer function, the proof works with these concepts and statements concretely (by manipulating them mathematically) to achieve the result. Gauss invented the greatest integer function for his third proof of QR in 1808 for this very purpose. Since it was long and messy, it was simplified by Eisenstein in 1844.

How a proof is presented by the author is also important. In my online text, "A Mathematical Analysis of the Greatest Integer Function", I include a full treatment of QR. For Gauss's third proof, I used Dence. His book, "Elements of the Theory of Numbers" is an advanced text and I felt that his presentation of the proof contained gaps. While his presentation might be appropriate for his objectives and target audience, I felt that it would seem more like an outline to undergraduate students. For this reason, when rewriting the proof, I took three measures to provide clarity. I put a lot of effort into closing every gap and providing a thorough explanation for certain parts, which accounts for a lot of its length. Since the proof has many components even after modifications by Eisenstein, I organized it by dividing it into sub-objectives and providing sub-proofs for each. I also include all three lattice-point diagrams with a numerical example in full detail. So, if you want to savor Gauss's third proof with modifications by Eisenstein, then you want to read my online text at www.greatestintegerfunctionresearch.org. My online text is also an excellent resource for various topics in number theory and analysis and is chockfull of original research.

I recommend Gauss's third proof with modifications by Eisenstein. In my opinion, it is by far the clearest and most straight-forward proof of Quadratic Reciprocity even though it is not the shortest. The proof makes no use of any mathematical discipline other than elementary number theory. That is, it uses no abstract algebra or combinatorics. Its charm is that it uses the greatest integer function in an advanced way in place of any advanced theory. The proof uses the greatest integer function to express a variety of number-theoretic concepts and statements involving integers mathematically. Then, using properties of the greatest integer function, the proof works with these concepts and statements concretely (by manipulating them mathematically) to achieve the result. Gauss invented the greatest integer function for his third proof of QR in 1808 for this very purpose. Since it was long and messy, it was simplified by Eisenstein in 1844.

How a proof is presented by the author is also important. In my online text, "A Mathematical Analysis of the Greatest Integer Function", I include a full treatment of QR. For Gauss's third proof, I used Dence. His book, "Elements of the Theory of Numbers" is an advanced text and I felt that his presentation of the proof contained gaps. While his presentation might be appropriate for his objectives and target audience, I felt that it would seem more like an outline to undergraduate students. For this reason, when rewriting the proof, I took three measures to provide clarity. I put a lot of effort into closing every gap and providing a thorough explanation for certain parts, which accounts for a lot of its length. Since the proof has many components even after modifications by Eisenstein, I organized it by dividing it into sub-objectives and providing sub-proofs for each. I also include all three lattice-point diagrams with a numerical example in full detail. So, if you want to savor Gauss's third proof with modifications by Eisenstein, then you want to read my online text at www.greatestintegerfunctionresearch.org (Wayback Machine. My online text is also an excellent resource for various topics in number theory and analysis and is chockfull of original research.