Mathematics of MOS: Difference between revisions
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{{nowrap|s {{=}} ''eg'' + ''fp''}} | {{nowrap|s {{=}} ''eg'' + ''fp''}} | ||
for appropriate integers ''c, d, e, f'', where {{nowrap|{{ | for appropriate integers ''c, d, e, f'', where {{nowrap|{{abs|''c''}}, {{abs|''e''}} < ''n''}}. | ||
Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} = ''p''. Since {{nowrap|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' − 1{{)}}}}. Hence we must have {{nowrap|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap|''c'' {{=}} ±''b''}} and {{nowrap|''e'' {{=}} ∓''a''}}, from the assumption that ''a'' and ''b'' are coprime. | Now we assume that ''g'' and ''p'' are linearly independent. By assumption {{nowrap|''a''L + ''b''s {{=}} (''ac'' + ''be'')''g'' + (''ad'' + ''bf'')''p''}} = ''p''. Since {{nowrap|''a''L + ''b''s}} occurs on the "brightest" mode, from generatedness we have {{nowrap|''ac'' + ''be'' ∈ {{(}}0, …, ''n'' − 1{{)}}}}. Hence we must have {{nowrap|''ac'' + ''be'' {{=}} 0}}, and thus {{nowrap|''c'' {{=}} ±''b''}} and {{nowrap|''e'' {{=}} ∓''a''}}, from the assumption that ''a'' and ''b'' are coprime. | ||