Almost Universal Quadratic Forms
An integral-valued positive definite quadratic form is called an almost universal quadratic form
when the form represents almost all positive integers.
We say the form is of deficit r when the cardinality of the set of numbers which are not represented by the form is r.
For example, the form x^2+y^2+z^2+10w^2 is almost universal of deficit 1 (The only 7 is not represented by the form).
The following Table shows all almost universal diagonal quaternary quadratic forms of deficit 0,1,2,3 (hence,
contains famous Ramanujan's 54 universal forms.)
Deficit 0
[ 1 1 1 1 ], [ 1 1 1 2 ], [ 1 1 1 3 ], [ 1 1 1 4 ], [ 1 1 1 5 ],
[ 1 1 1 6 ], [ 1 1 1 7 ], [ 1 1 2 10 ], [ 1 1 2 11 ], [ 1 1 2 12 ],
[ 1 1 2 13 ], [ 1 1 2 14 ], [ 1 1 2 2 ], [ 1 1 2 3 ], [ 1 1 2 4 ],
[ 1 1 2 5 ], [ 1 1 2 6 ], [ 1 1 2 7 ], [ 1 1 2 8 ], [ 1 1 2 9 ],
[ 1 1 3 3 ], [ 1 1 3 4 ], [ 1 1 3 5 ], [ 1 1 3 6 ], [ 1 2 2 2 ],
[ 1 2 2 3 ], [ 1 2 2 4 ], [ 1 2 2 5 ], [ 1 2 2 6 ], [ 1 2 2 7 ],
[ 1 2 3 10 ], [ 1 2 3 3 ], [ 1 2 3 4 ], [ 1 2 3 5 ], [ 1 2 3 6 ],
[ 1 2 3 7 ], [ 1 2 3 8 ], [ 1 2 3 9 ], [ 1 2 4 10 ], [ 1 2 4 11 ],
[ 1 2 4 12 ], [ 1 2 4 13 ], [ 1 2 4 14 ], [ 1 2 4 4 ], [ 1 2 4 5 ],
[ 1 2 4 6 ], [ 1 2 4 7 ], [ 1 2 4 8 ], [ 1 2 4 9 ], [ 1 2 5 10 ],
[ 1 2 5 6 ], [ 1 2 5 7 ], [ 1 2 5 8 ], [ 1 2 5 9 ]
Deficit 1
[ 1 1 1 10 ] {7}, [ 1 1 1 12 ] {7}, [ 1 1 1 14 ] {7}, [ 1 1 1 15 ] {7},
[ 1 1 1 9 ] {7}, [ 1 1 2 15 ] {14}, [ 1 1 2 17 ] {14}, [ 1 1 2 18 ] {14},
[ 1 1 2 19 ] {14}, [ 1 1 2 20 ] {14}, [ 1 1 2 21 ] {14}, [ 1 1 2 23 ] {14},
[ 1 1 2 24 ] {14}, [ 1 1 2 25 ] {14}, [ 1 1 2 27 ] {14}, [ 1 1 2 28 ] {14},
[ 1 1 2 29 ] {14}, [ 1 1 2 30 ] {14}, [ 1 1 3 10 ] {6}, [ 1 1 3 11 ] {6},
[ 1 1 3 13 ] {6}, [ 1 1 3 14 ] {6}, [ 1 1 3 15 ] {6}, [ 1 1 3 7 ] {6},
[ 1 1 3 8 ] {6}, [ 1 1 4 5 ] {3}, [ 1 1 4 6 ] {3}, [ 1 1 5 10 ] {3},
[ 1 1 5 11 ] {3}, [ 1 1 5 5 ] {3}, [ 1 1 5 6 ] {3}, [ 1 1 6 10 ] {3},
[ 1 1 6 11 ] {3}, [ 1 1 6 7 ] {3}, [ 1 1 6 8 ] {3}, [ 1 2 2 10 ] {7},
[ 1 2 2 12 ] {7}, [ 1 2 2 14 ] {7}, [ 1 2 2 15 ] {7}, [ 1 2 2 9 ] {7},
[ 1 2 3 11 ] {10}, [ 1 2 3 12 ] {10}, [ 1 2 3 13 ] {10}, [ 1 2 3 15 ] {10},
[ 1 2 3 17 ] {10}, [ 1 2 3 19 ] {10}, [ 1 2 3 20 ] {10}, [ 1 2 3 21 ] {10},
[ 1 2 3 22 ] {10}, [ 1 2 3 23 ] {10}, [ 1 2 3 24 ] {10}, [ 1 2 3 25 ] {10},
[ 1 2 3 26 ] {10}, [ 1 2 4 15 ] {14}, [ 1 2 4 17 ] {14}, [ 1 2 4 18 ] {14},
[ 1 2 4 19 ] {14}, [ 1 2 4 20 ] {14}, [ 1 2 4 21 ] {14}, [ 1 2 4 23 ] {14},
[ 1 2 4 24 ] {14}, [ 1 2 4 25 ] {14}, [ 1 2 4 27 ] {14}, [ 1 2 4 28 ] {14},
[ 1 2 4 29 ] {14}, [ 1 2 4 30 ] {14}, [ 1 2 5 11 ] {10}, [ 1 2 5 12 ] {10},
[ 1 2 5 13 ] {10}, [ 1 2 5 14 ] {10}, [ 1 2 5 5 ] {15}, [ 1 2 6 10 ] {5},
[ 1 2 6 11 ] {5}, [ 1 2 6 12 ] {5}, [ 1 2 6 13 ] {5}, [ 1 2 6 6 ] {5},
[ 1 2 7 10 ] {5}, [ 1 2 7 11 ] {5}, [ 1 2 7 12 ] {5}, [ 1 2 7 13 ] {5},
[ 1 2 7 8 ] {5}, [ 1 3 3 5 ] {2}, [ 1 3 5 6 ] {2}
Deficit 2
[ 1 1 1 11 ] {7.39}, [ 1 1 1 13 ] {7,28}, [ 1 1 1 17 ] {7,15},
[ 1 1 1 18 ] {7,15}, [ 1 1 1 20 ] {7,15}, [ 1 1 1 22 ] {7,15},
[ 1 1 1 23 ] {7,15}, [ 1 1 2 22 ] {14,78}, [ 1 1 2 26 ] {14,56},
[ 1 1 2 31 ] {14,30}, [ 1 1 2 33 ] {14,30}, [ 1 1 2 34 ] {14,30},
[ 1 1 2 35 ] {14,30}, [ 1 1 2 36 ] {14,30}, [ 1 1 2 37 ] {14,30},
[ 1 1 2 39 ] {14,30}, [ 1 1 2 40 ] {14,30}, [ 1 1 2 41 ] {14,30},
[ 1 1 2 43 ] {14,30}, [ 1 1 2 44 ] {14,30}, [ 1 1 2 45 ] {14,30},
[ 1 1 2 46 ] {14,30}, [ 1 1 3 16 ] {6,15}, [ 1 1 3 17 ] {6,15},
[ 1 1 3 19 ] {6,15}, [ 1 1 3 20 ] {6,15}, [ 1 1 3 22 ] {6,15},
[ 1 1 3 23 ] {6,15}, [ 1 1 4 10 ] {3,7}, [ 1 1 4 7 ] {3,35},
[ 1 1 5 12 ] {3,11}, [ 1 1 5 7 ] {3,19}, [ 1 1 5 9 ] {3,12},
[ 1 1 6 12 ] {3,39}, [ 1 1 6 13 ] {3,12}, [ 1 1 6 14 ] {3,12},
[ 1 1 6 16 ] {3,12}, [ 1 1 6 17 ] {3,12}, [ 1 1 6 19 ] {3,12},
[ 1 1 6 20 ] {3,12}, [ 1 2 10 12 ] {5,7}, [ 1 2 2 11 ] {7,39},
[ 1 2 2 13 ] {7,28}, [ 1 2 2 17 ] {7,15}, [ 1 2 2 18 ] {7,15},
[ 1 2 2 20 ] {7,15}, [ 1 2 2 22 ] {7,15}, [ 1 2 2 23 ] {7,15},
[ 1 2 3 14 ] {10,40}, [ 1 2 3 18 ] {10,58}, [ 1 2 3 27 ] {10,26},
[ 1 2 3 28 ] {10,26}, [ 1 2 3 29 ] {10,26}, [ 1 2 3 31 ] {10,26},
[ 1 2 3 33 ] {10,26}, [ 1 2 3 35 ] {10,26}, [ 1 2 3 36 ] {10,26},
[ 1 2 3 37 ] {10,26}, [ 1 2 3 38 ] {10,26}, [ 1 2 3 39 ] {10,26},
[ 1 2 3 40 ] {10,26}, [ 1 2 4 22 ] {14,78}, [ 1 2 4 26 ] {14,56},
[ 1 2 4 31 ] {14,30}, [ 1 2 4 33 ] {14,30}, [ 1 2 4 34 ] {14,30},
[ 1 2 4 35 ] {14,30}, [ 1 2 4 36 ] {14,30}, [ 1 2 4 37 ] {14,30},
[ 1 2 4 39 ] {14,30}, [ 1 2 4 40 ] {14,30}, [ 1 2 4 41 ] {14,30},
[ 1 2 4 43 ] {14,30}, [ 1 2 4 44 ] {14,30}, [ 1 2 4 45 ] {14,30},
[ 1 2 4 46 ] {14,30}, [ 1 2 5 15 ] {10,250}, [ 1 2 5 16 ] {10,15},
[ 1 2 5 17 ] {10,15}, [ 1 2 5 18 ] {10,15}, [ 1 2 5 19 ] {10,15},
[ 1 2 5 21 ] {10,15}, [ 1 2 5 22 ] {10,15}, [ 1 2 5 23 ] {10,15},
[ 1 2 5 24 ] {10,15}, [ 1 2 5 26 ] {10,15}, [ 1 2 5 27 ] {10,15},
[ 1 2 5 28 ] {10,15}, [ 1 2 5 29 ] {10,15}, [ 1 2 5 31 ] {10,15},
[ 1 2 5 32 ] {10,15}, [ 1 2 5 33 ] {10,15}, [ 1 2 5 34 ] {10,15},
[ 1 2 6 14 ] {5,13}, [ 1 2 6 18 ] {5,13}, [ 1 2 6 19 ] {5,13},
[ 1 2 6 20 ] {5,13}, [ 1 2 6 7 ] {5,20}, [ 1 2 6 9 ] {5,29},
[ 1 2 7 17 ] {5,14}, [ 1 2 7 18 ] {5,14}, [ 1 2 7 19 ] {5,14},
[ 1 2 7 20 ] {5,14}, [ 1 2 7 9 ] {5,14}, [ 1 2 8 12 ] {5,7},
[ 1 2 8 9 ] {5,7}, [ 1 2 9 10 ] {5,7}, [ 1 2 9 11 ] {5,7},
[ 1 2 9 13 ] {5,7}, [ 1 2 9 14 ] {5,7}, [ 1 3 4 6 ] {2,30},
[ 1 3 5 7 ] {2,22}, [ 1 3 5 8 ] {2,10}, [ 1 4 5 6 ] {2,3},
[ 1 4 6 11 ] {2,3}, [ 1 4 6 7 ] {2,3}, [ 1 4 6 8 ] {2,3},
[ 1 5 6 7 ] {2,3}, [ 1 5 6 8 ] {2,3}
Deficit 3
[ 1 1 1 19 ] {7,15,47}, [ 1 1 1 25 ] {7,15,23}, [ 1 1 1 26 ] {7,15,23},
[ 1 1 1 28 ] {7,15,23}, [ 1 1 10 13 ] {3,6,7}, [ 1 1 11 14 ] {3,6,7},
[ 1 1 2 38 ] {14,30,94}, [ 1 1 2 47 ] {14,30,46}, [ 1 1 2 49 ] {14,30,46},
[ 1 1 2 50 ] {14,30,46}, [ 1 1 2 51 ] {14,30,46}, [ 1 1 2 52 ] {14,30,46},
[ 1 1 2 53 ] {14,30,46}, [ 1 1 2 55 ] {14,30,46}, [ 1 1 2 56 ] {14,30,46},
[ 1 1 3 24 ] {6,15,78}, [ 1 1 3 25 ] {6,15,24}, [ 1 1 3 26 ] {6,15,24},
[ 1 1 3 28 ] {6,15,24}, [ 1 1 3 29 ] {6,15,24}, [ 1 1 3 31 ] {6,15,24},
[ 1 1 3 32 ] {6,15,24}, [ 1 1 4 14 ] {3,7,11}, [ 1 1 4 9 ] {3,7,28},
[ 1 1 5 14 ] {3,11,12}, [ 1 1 5 18 ] {3,11,12}, [ 1 1 6 21 ] {3,12,48},
[ 1 1 6 22 ] {3,12,21}, [ 1 1 6 23 ] {3,12,21}, [ 1 1 6 25 ] {3,12,21},
[ 1 1 6 26 ] {3,12,21}, [ 1 1 7 10 ] {3,6,31}, [ 1 1 7 12 ] {3,6,31},
[ 1 1 7 9 ] {3,6,31}, [ 1 1 8 10 ] {3,6,7}, [ 1 2 10 11 ] {5,7,31},
[ 1 2 10 13 ] {5,7,20}, [ 1 2 11 12 ] {5,7,10}, [ 1 2 11 15 ] {5,7,10},
[ 1 2 11 17 ] {5,7,10}, [ 1 2 11 20 ] {5,7,10}, [ 1 2 12 13 ] {5,7,10},
[ 1 2 12 15 ] {5,7,10}, [ 1 2 12 17 ] {5,7,10}, [ 1 2 12 20 ] {5,7,10},
[ 1 2 12 22 ] {5,7,10}, [ 1 2 12 24 ] {5,7,10}, [ 1 2 13 17 ] {5,7,10},
[ 1 2 13 20 ] {5,7,10}, [ 1 2 2 19 ] {7,15,47}, [ 1 2 2 25 ] {7,15,23},
[ 1 2 2 26 ] {7,15,23}, [ 1 2 2 28 ] {7,15,23}, [ 1 2 3 34 ] {10,26,74},
[ 1 2 3 41 ] {10,26,40}, [ 1 2 3 42 ] {10,26,40}, [ 1 2 4 38 ] {14,30,94},
[ 1 2 4 47 ] {14,30,46}, [ 1 2 4 49 ] {14,30,46}, [ 1 2 4 50 ] {14,30,46},
[ 1 2 4 51 ] {14,30,46}, [ 1 2 4 52 ] {14,30,46}, [ 1 2 4 53 ] {14,30,46},
[ 1 2 4 55 ] {14,30,46}, [ 1 2 4 56 ] {14,30,46}, [ 1 2 5 36 ] {10,15,35},
[ 1 2 5 37 ] {10,15,35}, [ 1 2 5 38 ] {10,15,35}, [ 1 2 5 39 ] {10,15,35},
[ 1 2 6 17 ] {5,13,37}, [ 1 2 6 21 ] {5,13,20}, [ 1 2 7 16 ] {5,14,21},
[ 1 2 8 11 ] {5,7,39}, [ 1 2 8 13 ] {5,7,28}, [ 1 2 9 12 ] {5,7,35},
[ 1 2 9 20 ] {5,7,14}, [ 1 3 3 8 ] {2,5,26}, [ 1 3 4 5 ] {2,15,38},
[ 1 3 5 11 ] {2,10,22}, [ 1 3 5 14 ] {2,10,11}, [ 1 3 5 9 ] {2,11,74},
[ 1 3 6 7 ] {2,5,30}, [ 1 3 6 8 ] {2,5,46}, [ 1 3 7 14 ] {2,5,6},
[ 1 4 4 6 ] {2,3,34}, [ 1 4 6 10 ] {2,3,12}, [ 1 4 6 17 ] {2,3,12},
[ 1 5 6 12 ] {2,3,8}
Last updated: 21 February 2001