Express each integer between 1 and 1000 as a sum of four cubes of integers (which can be positive, negative, or zero).
(In reply to
Computer findings, leaving 45 cases for analytic methods by Charlie)
605 = 0 + 1 + -125 + 729
606 = 1 + 1 + -125 + 729
607 = -8 + -1331 + -4913 + 6859
608 = -64 + 125 + -2197 + 2744
609 = 8 + -64 + -64 + 729
610 = 0 + 8 + -729 + 1331
611 = 8 + 27 + 64 + 512
612 = 0 + 8 + -125 + 729
613 = 27 + 27 + 216 + 343
614 = 0 + 343 + -729 + 1000
615 = 0 + -1331 + -4913 + 6859
616 = 1 + -1331 + -4913 + 6859
617 = -64 + -343 + 512 + 512
618 = 8 + 8 + -729 + 1331
619 = 8 + 64 + -2197 + 2744
620 = 27 + 125 + 125 + 343
621 = 64 + 125 + 216 + 216
622 = 0 + -64 + 343 + 343
623 = 0 + -8 + -2744 + 3375
624 = 1 + 64 + 216 + 343
625 = 64 + -125 + 343 + 343
missing 626 - 626
627 = 512 + 512 + 1331 + -1728
628 = -1 + -8 + 125 + 512
629 = 0 + -8 + 125 + 512
630 = 27 + 27 + 64 + 512
631 = 8 + 64 + 216 + 343
632 = 0 + 1 + -2744 + 3375
633 = 1 + 1 + -2744 + 3375
missing 634 - 634
635 = -1 + -1 + 125 + 512
636 = 0 + -1 + 125 + 512
637 = 0 + 0 + 125 + 512
638 = 0 + 1 + 125 + 512
639 = 1 + 1 + 125 + 512
640 = 0 + 64 + 64 + 512
641 = 1 + 64 + 64 + 512
642 = -8 + -216 + -1331 + 2197
missing 643 - 643
644 = -1 + 8 + 125 + 512
645 = 0 + 8 + 125 + 512
646 = 1 + 8 + 125 + 512
647 = 0 + -1000 + -1728 + 3375
648 = 8 + 64 + 64 + 512
649 = 1 + 216 + 216 + 216
650 = 27 + 64 + 216 + 343
651 = 1 + -216 + -1331 + 2197
missing 652 - 652
653 = 8 + 8 + 125 + 512
654 = -27 + -343 + 512 + 512
655 = 8 + -1000 + -1728 + 3375
656 = 8 + 216 + 216 + 216
657 = 64 + 125 + 125 + 343
658 = 0 + 1 + -343 + 1000
659 = 0 + -27 + 343 + 343
660 = 1 + -27 + 343 + 343
661 = -8 + 1000 + 1000 + -1331
662 = -1000 + -1000 + 1331 + 1331
663 = -1 + -1 + -64 + 729
664 = 0 + -1 + -64 + 729
665 = 1 + 27 + 125 + 512
666 = 0 + 1 + -64 + 729
667 = 27 + 64 + 64 + 512
668 = 0 + 64 + -125 + 729
669 = 0 + 1000 + 1000 + -1331
670 = 1 + 1000 + 1000 + -1331
671 = -1 + 125 + -2197 + 2744
672 = 8 + 27 + 125 + 512
673 = 0 + 8 + -64 + 729
674 = 0 + 343 + -1000 + 1331
675 = 27 + 216 + 216 + 216
676 = 1 + -27 + -27 + 729
677 = 8 + 1000 + 1000 + -1331
678 = 0 + -8 + 343 + 343
679 = 1 + -8 + 343 + 343
680 = 8 + 125 + -2197 + 2744
681 = 0 + -343 + 512 + 512
682 = 125 + 125 + 216 + 216
683 = 8 + -27 + -27 + 729
684 = 0 + 27 + -343 + 1000
685 = 1 + 125 + 216 + 343
686 = 0 + 0 + 343 + 343
687 = 64 + 64 + 216 + 343
688 = 1 + 1 + 343 + 343
689 = 8 + -343 + 512 + 512
690 = 216 + -343 + -4096 + 4913
691 = 27 + 27 + 125 + 512
692 = 8 + 125 + 216 + 343
693 = 1 + 27 + -64 + 729
694 = 0 + 8 + 343 + 343
695 = 1 + 8 + 343 + 343
696 = 1 + 64 + -2744 + 3375
697 = 343 + -512 + -1331 + 2197
698 = -64 + 125 + 125 + 512
699 = -1 + 1331 + 2744 + -3375
700 = 0 + 1331 + 2744 + -3375
701 = 0 + -1 + -27 + 729
702 = 1 + 64 + 125 + 512
703 = 0 + 1 + -27 + 729
704 = 64 + 64 + 64 + 512
705 = 1 + 216 + -512 + 1000
706 = -125 + 343 + -512 + 1000
707 = -1000 + -2197 + -4096 + 8000
708 = 8 + 1331 + 2744 + -3375
709 = 8 + 64 + 125 + 512
710 = 0 + 8 + -27 + 729
711 = 27 + 125 + 216 + 343
712 = 64 + 216 + 216 + 216
713 = 0 + -8 + -8 + 729
714 = 1 + 27 + 343 + 343
715 = -125 + -512 + -2744 + 4096
716 = -343 + 512 + -2197 + 2744
717 = -27 + -1000 + -1000 + 2744
718 = 125 + 125 + 125 + 343
719 = -1 + -1 + -8 + 729
720 = 0 + -1 + -8 + 729
721 = 8 + 27 + 343 + 343
722 = 0 + 1 + -8 + 729
723 = 1 + 1 + -8 + 729
missing 724 - 725
726 = -1 + -1 + -1 + 729
727 = 0 + -1 + -1 + 729
728 = 27 + 64 + 125 + 512
729 = 0 + 0 + 0 + 729
730 = 1 + 1 + 216 + 512
731 = 0 + 1 + 1 + 729
732 = 1 + 1 + 1 + 729
733 = -8 + -125 + -1331 + 2197
734 = -64 + -343 + -6859 + 8000
735 = 8 + 125 + -729 + 1331
736 = 0 + 8 + 216 + 512
737 = 1 + 8 + 216 + 512
738 = 0 + 1 + 8 + 729
739 = 1 + 1 + 8 + 729
740 = 27 + 27 + 343 + 343
741 = 0 + -125 + -1331 + 2197
742 = 1 + -125 + -1331 + 2197
743 = -1 + -1000 + -1000 + 2744
744 = 8 + 8 + 216 + 512
745 = 0 + 8 + 8 + 729
746 = 1 + 8 + 8 + 729
747 = 8 + -729 + -729 + 2197
748 = 64 + 125 + 216 + 343
749 = 1 + -8 + 27 + 729
750 = 0 + 64 + 343 + 343
751 = 1 + 64 + 343 + 343
752 = 8 + -1000 + -1000 + 2744
753 = 8 + 8 + 8 + 729
754 = 1 + -64 + -4096 + 4913
755 = 0 + -1 + 27 + 729
756 = 1 + 27 + 216 + 512
757 = 0 + 1 + 27 + 729
758 = 1 + 1 + 27 + 729
759 = 64 + 64 + -2744 + 3375
missing 760 - 760
761 = 8 + -64 + -4096 + 4913
762 = 0 + 125 + 125 + 512
763 = 1 + 125 + 125 + 512
764 = 0 + 8 + 27 + 729
765 = 1 + 8 + 27 + 729
766 = 0 + -27 + 64 + 729
767 = 1 + -27 + 64 + 729
768 = -8 + -8 + -216 + 1000
missing 769 - 769
770 = 8 + 125 + 125 + 512
771 = 8 + 216 + -2197 + 2744
772 = 8 + 8 + 27 + 729
773 = 125 + 216 + 216 + 216
774 = 8 + -27 + 64 + 729
775 = 0 + 216 + 216 + 343
776 = 1 + 216 + 216 + 343
777 = 27 + 64 + 343 + 343
missing 778 - 778
779 = -64 + 512 + -1000 + 1331
780 = 8 + -216 + -343 + 1331
781 = -1 + 125 + -343 + 1000
782 = 27 + 27 + 216 + 512
783 = 8 + 216 + 216 + 343
784 = 1 + 27 + 27 + 729
785 = 0 + 1 + -216 + 1000
786 = 1 + 1 + -216 + 1000
missing 787 - 788
789 = 27 + 125 + 125 + 512
790 = 0 + -27 + -4096 + 4913
791 = 8 + 27 + 27 + 729
792 = 0 + 8 + -216 + 1000
793 = 1 + 64 + 216 + 512
794 = 0 + 1 + 64 + 729
795 = 1 + 1 + 64 + 729
796 = 343 + 1000 + 2197 + -2744
797 = -1 + -343 + -6859 + 8000
798 = 0 + -343 + -6859 + 8000
799 = 1 + -343 + -6859 + 8000
800 = 8 + 64 + 216 + 512
801 = 0 + 8 + 64 + 729
802 = 1 + 8 + 64 + 729
803 = 1 + -64 + -1331 + 2197
804 = -8 + 343 + -1728 + 2197
805 = 64 + -125 + -1331 + 2197
806 = 8 + -343 + -6859 + 8000
807 = -1 + -216 + 512 + 512
808 = 0 + -216 + 512 + 512
809 = 8 + 8 + 64 + 729
810 = 27 + 27 + 27 + 729
811 = 0 + -8 + -512 + 1331
812 = 1 + 125 + 343 + 343
813 = 1 + 343 + -1728 + 2197
814 = 64 + 64 + 343 + 343
815 = -1 + -1 + -4096 + 4913
816 = 0 + -1 + -4096 + 4913
817 = 0 + 0 + -4096 + 4913
818 = 0 + 1 + -4096 + 4913
819 = 8 + 125 + 343 + 343
820 = 0 + 1 + -512 + 1331
821 = 1 + 27 + 64 + 729
822 = -125 + -125 + 343 + 729
823 = -8 + 343 + -512 + 1000
824 = -1 + 8 + -4096 + 4913
825 = 0 + 8 + -4096 + 4913
826 = 64 + 125 + 125 + 512
827 = 0 + 8 + -512 + 1331
828 = 8 + 27 + 64 + 729
829 = 1 + -27 + 343 + 512
830 = -1 + 343 + -512 + 1000
831 = 0 + 343 + -512 + 1000
832 = 1 + 343 + -512 + 1000
833 = 8 + 8 + -4096 + 4913
834 = -1 + -343 + -2197 + 3375
835 = 0 + -343 + -2197 + 3375
836 = 1 + -343 + -2197 + 3375
837 = 0 + -1331 + -5832 + 8000
838 = 27 + 125 + 343 + 343
839 = 64 + 216 + 216 + 343
840 = 0 + -512 + -2744 + 4096
841 = 1 + -512 + -2744 + 4096
842 = -1 + 512 + -1000 + 1331
843 = 0 + 512 + -1000 + 1331
844 = 0 + 27 + -4096 + 4913
845 = 0 + 2197 + 2744 + -4096
846 = 0 + -8 + 125 + 729
847 = 27 + 27 + 64 + 729
848 = 0 + 64 + -216 + 1000
849 = 1 + 64 + -216 + 1000
850 = -8 + -8 + -1331 + 2197
851 = 8 + 512 + -1000 + 1331
852 = 8 + 27 + -4096 + 4913
853 = 0 + -1 + 125 + 729
854 = 1 + 125 + 216 + 512
855 = 0 + 0 + 343 + 512
856 = 1 + 1 + 125 + 729
857 = 1 + 1 + 343 + 512
858 = 1 + 64 + 64 + 729
859 = 1 + -8 + -1331 + 2197
860 = -64 + -64 + -343 + 1331
861 = 8 + 125 + 216 + 512
862 = 0 + 8 + 125 + 729
863 = 1 + 8 + 125 + 729
864 = 1 + 8 + 343 + 512
865 = 8 + 64 + 64 + 729
866 = 0 + 0 + -1331 + 2197
867 = 0 + 1 + -1331 + 2197
868 = 1 + 1 + -1331 + 2197
869 = 125 + -1000 + -1000 + 2744
870 = 8 + 8 + 125 + 729
871 = 8 + 8 + 343 + 512
872 = 0 + -64 + -64 + 1000
873 = 0 + 216 + -343 + 1000
874 = 0 + 8 + -1331 + 2197
875 = 64 + 125 + 343 + 343
876 = 0 + 1 + -125 + 1000
877 = 1 + 1 + -125 + 1000
878 = -64 + 125 + -4096 + 4913
missing 879 - 879
880 = 27 + 125 + 216 + 512
881 = 0 + 27 + 125 + 729
882 = 1 + 27 + 125 + 729
883 = 1 + 27 + 343 + 512
884 = 27 + 64 + 64 + 729
885 = -8 + 27 + -1331 + 2197
missing 886 - 886
887 = 125 + 125 + 125 + 512
888 = -64 + -64 + -1728 + 2744
889 = 8 + 27 + 125 + 729
890 = 8 + 27 + 343 + 512
891 = 0 + -125 + -1728 + 2744
892 = 0 + -27 + -4913 + 5832
893 = 0 + 27 + -1331 + 2197
894 = 1 + 27 + -1331 + 2197
895 = 64 + 343 + -512 + 1000
896 = -64 + -64 + 512 + 512
897 = -1 + -343 + 512 + 729
898 = 0 + -343 + 512 + 729
899 = 0 + -125 + 512 + 512
900 = 125 + 216 + 216 + 343
901 = 0 + -512 + -1331 + 2744
902 = 0 + 27 + -125 + 1000
903 = 1 + 216 + 343 + 343
904 = 64 + -512 + -2744 + 4096
missing 905 - 905
906 = 8 + -343 + 512 + 729
907 = 8 + -125 + 512 + 512
908 = 27 + 27 + 125 + 729
909 = 27 + 27 + 343 + 512
910 = 8 + 216 + 343 + 343
911 = 0 + -8 + -4913 + 5832
912 = 1 + -8 + -4913 + 5832
missing 913 - 914
915 = -27 + 125 + -4096 + 4913
916 = -8 + -64 + -343 + 1331
917 = 64 + 125 + 216 + 512
918 = 0 + -1 + -4913 + 5832
919 = 1 + 64 + 125 + 729
920 = 1 + 64 + 343 + 512
921 = 64 + 64 + 64 + 729
922 = -8 + 64 + -1331 + 2197
923 = -1 + -64 + -343 + 1331
924 = 0 + -64 + -343 + 1331
925 = 0 + -216 + -6859 + 8000
926 = 8 + 64 + 125 + 729
927 = 8 + 64 + 343 + 512
928 = 0 + -8 + -64 + 1000
929 = 27 + 216 + 343 + 343
930 = 0 + 64 + -1331 + 2197
931 = 1 + 64 + -1331 + 2197
932 = 8 + -64 + -343 + 1331
933 = 8 + -216 + -6859 + 8000
934 = 0 + 1331 + 1331 + -1728
935 = 0 + -1 + -64 + 1000
936 = 125 + 125 + 343 + 343
937 = 0 + 1 + -64 + 1000
938 = 1 + 1 + -64 + 1000
939 = 0 + 64 + -125 + 1000
940 = 1 + 64 + -125 + 1000
941 = -1 + 125 + -4096 + 4913
942 = 0 + 125 + -4096 + 4913
943 = 1 + 125 + -4096 + 4913
944 = 0 + 8 + -64 + 1000
945 = 1 + 216 + 216 + 512
946 = 27 + 64 + 343 + 512
947 = 1 + 1 + 216 + 729
948 = 1 + -125 + 343 + 729
missing 949 - 949
950 = 8 + 125 + -4096 + 4913
951 = -1 + -64 + -1728 + 2744
952 = 8 + 216 + 216 + 512
953 = 0 + 8 + 216 + 729
954 = 1 + 8 + 216 + 729
955 = 8 + -125 + 343 + 729
956 = 0 + -512 + -729 + 2197
957 = 1 + -512 + -729 + 2197
958 = 343 + -1331 + -4913 + 6859
959 = -1 + -64 + 512 + 512
960 = 0 + -64 + 512 + 512
961 = 8 + 8 + 216 + 729
962 = 0 + -216 + -2197 + 3375
963 = 0 + 27 + -64 + 1000
964 = 1 + 27 + -64 + 1000
965 = 0 + -8 + -27 + 1000
966 = 64 + 216 + 343 + 343
missing 967 - 967
968 = 8 + -64 + 512 + 512
969 = 8 + -27 + -343 + 1331
970 = 8 + -216 + -2197 + 3375
971 = 27 + 216 + 216 + 512
972 = 0 + -1 + -27 + 1000
973 = 1 + 27 + 216 + 729
974 = 0 + 1 + -27 + 1000
975 = 1 + 1 + -27 + 1000
976 = -8 + -8 + -8 + 1000
977 = 512 + 1331 + 1331 + -2197
978 = 125 + 125 + 216 + 512
979 = 0 + 125 + 125 + 729
980 = 1 + 125 + 125 + 729
981 = 1 + 125 + 343 + 512
982 = 64 + 64 + 125 + 729
983 = 64 + 64 + 343 + 512
984 = 0 + -8 + -8 + 1000
985 = 1 + -8 + -8 + 1000
986 = -1 + -1 + -343 + 1331
987 = 8 + 125 + 125 + 729
988 = 8 + 125 + 343 + 512
989 = 0 + 1 + -343 + 1331
990 = 0 + -125 + -216 + 1331
991 = 216 + 216 + 216 + 343
992 = 0 + 0 + -8 + 1000
993 = 0 + 1 + -8 + 1000
994 = 1 + 1 + -8 + 1000
995 = -1 + 8 + -343 + 1331
996 = 0 + 8 + -343 + 1331
997 = 0 + -27 + 512 + 512
998 = 0 + -1 + -1 + 1000
999 = 27 + 27 + 216 + 729
1000 = 0 + 0 + 0 + 1000
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Posted by Charlie
on 2013-10-23 12:24:51 |
re: Computer findings, leaving 45 cases for analytic methods--list continued
