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SOLUTION OF THE PROBLEM A definitive answer to the major question was made in 1962 by Frank Adams . He showed in all dimensions ''N'' that the conjectural number ρ(''N'') of linearly-independent Vector Field s on the (''N − 1'')-sphere in ''N''-dimensional Euclidean Space was correct. It was already known, by direct construction, that there were such fields; Adams applied Homotopy Theory to prove that no more independent vector fields could be found. TECHNICAL DETAILS In detail, the question applies to the 'round spheres' (not Exotic Sphere s); and to their Tangent Bundle s. The case of ''N'' odd is taken care of by the PoincarĂ©-Hopf Index Theorem (see Hairy Ball Theorem ), so the case ''N'' even is an extension of that. The maximum number of continuous (''smooth'' would be no different here) pointwise linearly-independent vector fields on the ''(N − 1)''-sphere is computable by this formula: write ''N'' as the product of an odd number ''A'' and a Power Of Two 2''B''. Write B Then :ρ(''N'') = 2''c'' + 8''d'' − 1. The construction of the fields is related to the real Clifford Algebra s, which is a theory with a periodicity ''modulo'' 8 that also shows up here. By the Gram-Schmidt Process , it is the same to ask for (pointwise) linear independence or fields that give an Orthonormal Basis at each point. RADON-HURWITZ NUMBERS The numbers ρ(''n'') are the Radon-Hurwitz numbers, so-called from the earlier work of Johann Radon (1922) and Adolf Hurwitz (1923) in this area. A Recurrence Relation is easy to give. These numbers occur also in other, related areas. In Matrix Theory , the Radon-Hurwitz number counts the maximum size of a linear subspace of the real ''n''×''n'' matrices, for which each non-zero matrix is a Similarity , i.e. a product of an Orthogonal Matrix and a Scalar Matrix . The classical results were revisited in 1952 by Beno Eckmann . They are now applied in areas including Coding Theory and Theoretical Physics . REFERENCE
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