How to interpret Fueter's description of Diophantine equations with no rational points?

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I'm an amateur mathematician so please forgive me if this is a basic question, but could you help me understand the meaning of the following snippet from a journal article (referring to work of Fueter):

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Isn't 2mod9 equal to 2? How does this allow me to calculate a collection of delta k values? In the second statement, is it (8mod16 or 12mod16) or is it (8 or 12mod16? What does the 3 cross hk mean?

Any help you can provide would be greatly appreciated. Thanks!