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Given the positive integers m,n, solving the well known symmetric Diophantine equation xm+kyn=zm+kwn, where k is a rational number, is a challenge. By computer calculations, we show that for all integers k from 1 to 500, the Diophantine equation x6+ky3=z6+kw3 has infinitely many nontrivial (y≠w) rational solutions. Clearly, the same result holds for positive integers k whose cube-free part is n...

Critical Load (CL) methodology is currently used throughout Europe to assess the risks of ecological damage due to sulphur and nitrogen emissions. Critical acid neutralising capacity (ANC CRIT) is used in CL estimates for freshwater systems as a surrogate for biological damage. Although UK CL maps presently use an ANC value of 0 meq l –1 , this value has been based largely on Norwegian lake stu...

6-carbethoxy-5-(3'-bromophenyl)-3-aryl-2-cyclohexenones 2a-j were obtained from the1-aryl-3-(3'-bromophenyl)-2-propene-1-ones 1a-j by micheal addition of ethyl acetoacetate, followed by internal claisen condensation. reaction of 2a-j with hydrazine hydrate afforded the corresponding 6-aryl-4-(3'-bromophenyl)-3-oxo-2,3a,4,5-tetrahydro-2h-indazoles 3a-j. the structures of newly syn...