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English version: Journal of Applied and Industrial Mathematics, 2017, 11:3, 334346 

Volume 24, No 3, 2017, P. 104124 UDC 519.6
Keywords: Frobenius number, primitive set, additive semigroup, computational complexity. DOI: 10.17377/daio.2017.24.537 Vladimir M. Fomichev ^{1,2} Received 8 April 2016 References[1] V. I. Arnold, Experimental observation of mathematical facts, MTsNMO, Moscow, 2006.[2] A. V. Ustinov, The solution of Arnold’s problem on the weak asymptotics of Frobenius numbers with three arguments, Mat. Sb., 200, No. 4, 131–160, 2009. Translated in Sb. Math., 200, No. 4, 597–627, 2009. [3] A. V. Ustinov, Geometric proof of Rødseth’s formula for Frobenius numbers, Tr. Mat. Inst. Steklova, 276, 280–287, 2012. Translated in Proc. Steklov Inst. Math., 276, No. 1, 275–282, 2012. [4] V. M. Fomichev, Primitive sets of numbers being equivalent by Frobenius, Prikladn. Diskretn. Mat., No. 1, 20–26, 2014. [5] V. M. Fomichev, Estimates for exponent of some graphs by means of Frobenius’s numbers of three arguments, Prikladn. Diskretn. Mat., No. 2, 88–96, 2014. [6] S. Böcker and Zs. Lipták, The Money Changing Problem revisited: Computing the Frobenius number in time $O(ka_1)$, in Computing and Combinatorics (Proc. 4th Annu. Int. Conf., Kunming, China, Aug. 16–19, 2005), pp. 965–974, Springer, Heidelberg, 2005 (Lect. Notes Comput. Sci., Vol. 3595). [7] C. Bogart, Calculating Frobenius numbers with Boolean Toeplitz matrix multiplication, 2009. Available at http://quetzal.bogarthome.net/frobenius.pdf (accessed Mar. 10, 2017). [8] A. Brauer, On a problem of partitions, Am. J. Math., 64, 299–312, 1942. [9] A. Brauer and J. E. Shockley, On a problem of Frobenius, J. Reine Angew. Math., 211, 215–220, 1962. [10] F. Curtis, On formulas for the Frobenius number of a numerical semigroup, Math. Scand., 67, 190–192, 1990. [11] B. R. Heap and M. S. Lynn, A graphtheoretic algorithm for the solution of a linear Diophantine problem of Frobenius, Numer. Math., 6, 346–354, 1964. [12] B. R. Heap and M. S. Lynn, On a linear Diophantine problem of Frobenius: An improved algorithm, Numer. Math., 7, 226–231, 1965. [13] D. B. Johnson, Efficient algorithms for shortest paths in space networks, J. ACM, 24, 1–13, 1977. [14] M. Nijenhuis, A minimalpath algorithm for the “Money changing problem”, Am. Math. Mon., 86, 832–835, 1979. [15] J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Clarendon Press, Oxford, 2005 (Oxf. Lect. Ser. Math. Appl., Vol. 30). [16] J. J. Sylvester, Problem 7382, in Mathematical Questions with Their Solutions: From the “Educational Times”, Vol. 41, p. 21, Francis Hodgson, London, 1884. Available at http://archive.org/stream/mathematicalque05unkngoog#page/n150/mode/2up. Accessed Mar. 10, 2017. 

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