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In this paper, we will prove there are infinitely many integers n such that n 2 − 1 is square-free and Q( √ n 2 − 1) admits universal octonary diagonal quadratic forms. (2000) . Primary 11E12, Secondary 11E20. Mathematics Subject Classificationdoi:10.1007/s000140050133 fatcat:a7nh26k66fec5eksn4gpeka6ii