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Voici maintenant la liste des entiers de 1 \[AGrave] 100 non somme de 3 carr\ \[EAcute]s :\ \>", "Text", CellChangeTimes->{{3.460133745390625*^9, 3.4601337863125*^9}, { 3.46901552696875*^9, 3.46901554134375*^9}, {3.469798125203125*^9, 3.469798132609375*^9}, 3.4698006934375*^9}], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{ RowBox[{"n", "=", "100"}], ";", RowBox[{"A", "=", RowBox[{"Range", "[", RowBox[{"0", ",", "n"}], "]"}]}], ";", RowBox[{"y", "=", RowBox[{"x", "-", RowBox[{"a", "^", "2"}]}]}], ";"}], "\[IndentingNewLine]", RowBox[{"Do", "[", "\[IndentingNewLine]", RowBox[{ RowBox[{"Do", "[", RowBox[{ RowBox[{ RowBox[{"y", "=", RowBox[{"x", "-", RowBox[{"a", "^", "2"}]}]}], ";", "\[IndentingNewLine]", RowBox[{"Do", "[", "\[IndentingNewLine]", " ", RowBox[{ RowBox[{ RowBox[{"z", "=", RowBox[{"y", "-", RowBox[{"b", "^", "2"}]}]}], ";", RowBox[{"c", "=", RowBox[{"Floor", "[", RowBox[{"Sqrt", "[", "z", "]"}], "]"}]}], ";", RowBox[{"If", "[", " ", RowBox[{ RowBox[{ RowBox[{"c", "^", "2"}], "\[Equal]", "z"}], ",", " ", RowBox[{"A", "=", RowBox[{"Complement", "[", RowBox[{"A", ",", RowBox[{"{", "x", "}"}]}], "]"}]}]}], " ", "]"}]}], "\[IndentingNewLine]", ",", RowBox[{"{", " ", RowBox[{"b", ",", "a", ",", RowBox[{"Sqrt", "[", RowBox[{"y", "/", "2"}], "]"}]}], "}"}]}], "]"}]}], "\[IndentingNewLine]", ",", " ", RowBox[{"{", " ", RowBox[{"a", " ", ",", "0", " ", ",", RowBox[{"Sqrt", "[", RowBox[{"x", "/", "3"}], "]"}]}], "}"}]}], "]"}], "\[IndentingNewLine]", ",", RowBox[{"{", RowBox[{"x", ",", "0", ",", "n"}], "}"}]}], "]"}], "\[IndentingNewLine]", "A"}], "Input", CellChangeTimes->{{3.460133818984375*^9, 3.46013383559375*^9}, 3.46013388165625*^9, {3.460133930578125*^9, 3.460133940296875*^9}, { 3.460134282703125*^9, 3.460134292453125*^9}, {3.469797483234375*^9, 3.469797595484375*^9}, {3.4697976325625*^9, 3.46979765678125*^9}, { 3.469797768125*^9, 3.469797787484375*^9}, {3.4697978238125*^9, 3.46979782753125*^9}, {3.4697978896875*^9, 3.4697979005625*^9}, 3.469797955453125*^9, {3.469798143203125*^9, 3.469798160484375*^9}, { 3.469798205359375*^9, 3.469798205984375*^9}, {3.4698007361875*^9, 3.469800752359375*^9}, {3.501242994140625*^9, 3.50124300759375*^9}, { 3.501243137015625*^9, 3.501243171125*^9}, {3.50124323934375*^9, 3.501243256875*^9}, {3.50124329540625*^9, 3.501243296828125*^9}, { 3.5012433345625*^9, 3.501243336109375*^9}, {3.50124338128125*^9, 3.501243457203125*^9}, {3.501243491421875*^9, 3.50124354715625*^9}, { 3.5012438585625*^9, 3.501243916734375*^9}, {3.501244004140625*^9, 3.501244025984375*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "7", ",", "15", ",", "23", ",", "28", ",", "31", ",", "39", ",", "47", ",", "55", ",", "60", ",", "63", ",", "71", ",", "79", ",", "87", ",", "92", ",", "95"}], "}"}]], "Output", CellChangeTimes->{{3.469797808046875*^9, 3.4697978285*^9}, { 3.469797894984375*^9, 3.469797901828125*^9}, 3.46979795625*^9, { 3.469798144578125*^9, 3.46979816165625*^9}, 3.469798207265625*^9, { 3.4698007378125*^9, 3.46980075396875*^9}, 3.5012429748125*^9, 3.501243009265625*^9, 3.5012432081875*^9, 3.501243258125*^9, 3.501243297765625*^9, {3.50124339428125*^9, 3.50124345796875*^9}, { 3.501243492125*^9, 3.501243548*^9}, 3.501243852578125*^9, { 3.50124389390625*^9, 3.50124390853125*^9}, 3.501244030453125*^9}] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[{ RowBox[{"B", "=", RowBox[{"Select", "[", RowBox[{"A", ",", "OddQ"}], "]"}]}], "\[IndentingNewLine]", RowBox[{"CC", "=", RowBox[{"Select", "[", RowBox[{"A", ",", "EvenQ"}], "]"}]}]}], "Input", CellChangeTimes->{{3.46013399475*^9, 3.4601341873125*^9}, { 3.4601342995625*^9, 3.460134362921875*^9}}], Cell[BoxData[ RowBox[{"{", RowBox[{ "7", ",", "15", ",", "23", ",", "31", ",", "39", ",", "47", ",", "55", ",", "63", ",", "71", ",", "79", ",", "87", ",", "95"}], "}"}]], "Output", CellChangeTimes->{ 3.4601341886875*^9, {3.460134303921875*^9, 3.460134356578125*^9}, 3.469797602453125*^9, 3.469797928828125*^9, 3.469798210125*^9, 3.501244034859375*^9}], Cell[BoxData[ RowBox[{"{", RowBox[{"28", ",", "60", ",", "92"}], "}"}]], "Output", CellChangeTimes->{ 3.4601341886875*^9, {3.460134303921875*^9, 3.460134356578125*^9}, 3.469797602453125*^9, 3.469797928828125*^9, 3.469798210125*^9, 3.501244034859375*^9}] }, Open ]], Cell[TextData[{ "Je remarque que B est form\[EAcute] des entiers < 100 de la forme 8 k - 1 \ et que C est form\[EAcute] des produits de ces derniers par des puissances de \ 4.\nDonc je conjecture : n \[NotEqual]", Cell[BoxData[ RowBox[{ SuperscriptBox["a", "2"], "+", SuperscriptBox["b", "2"], "+", SuperscriptBox["c", "2"]}]], CellChangeTimes->{3.4667885629375*^9}], " ssi n est de la forme ", Cell[BoxData[ RowBox[{ SuperscriptBox["4", "q"], " ", RowBox[{"(", RowBox[{ RowBox[{"8", " ", "k"}], "-", "1"}], ")"}]}]], CellChangeTimes->{{3.46678859771875*^9, 3.46678861946875*^9}}], ". Ceci a \[EAcute]t\[EAcute] prouv\[EAcute]." }], "Text", CellChangeTimes->{ 3.46013439903125*^9, {3.460134433203125*^9, 3.460134482609375*^9}, 3.466788516140625*^9, 3.466788570921875*^9, {3.466788603828125*^9, 3.466788672375*^9}}] }, Open ]] }, WindowToolbars->"EditBar", WindowSize->{1272, 665}, WindowMargins->{{3, Automatic}, {-2, Automatic}}, PrintingCopies->1, PrintingPageRange->{Automatic, Automatic}, PrintingOptions->{"PrintingMargins"->{{54.000000584999995`, \ 54.000000584999995`}, {56.692913999999995`, 56.692913999999995`}}}, Magnification:>FEPrivate`If[ FEPrivate`Equal[FEPrivate`$VersionNumber, 6.], 1.5, 1.5 Inherited], FrontEndVersion->"7.0 for Microsoft Windows (32-bit) (February 18, 2009)", StyleDefinitions->"Default.nb" ] (* End of Notebook Content *) (* Internal cache information *) (*CellTagsOutline CellTagsIndex->{} *) (*CellTagsIndex CellTagsIndex->{} *) (*NotebookFileOutline Notebook[{ Cell[545, 20, 418, 9, 91, "Input"], Cell[CellGroupData[{ Cell[988, 33, 92, 1, 42, 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