How Fermat Factored Integers

July 18, 2014

The table of squares is calculated by the following program, using the printf function from a previous exercise:

; print table of squares to a million
(begin
  (define (square x) (* x x))
  (define (print-header)
    (printf "--- ------ ------ ------ ------ ------ ------ ------ ------ ------\n")
    (printf " 1 2 3 4 5 6 7 8 9\n")
    (printf "--- ------ ------ ------ ------ ------ ------ ------ ------ ------\n"))
  (do ((tens 0 (+ tens 1))) ((= tens 100))
    (when (zero? (modulo tens 10)) (print-header))
    (printf "%3d" (* tens 10))
    (do ((ones 1 (+ ones 1))) ((= ones 10))
      (printf " %6d" (square (+ (* tens 10) ones))))
    (printf "\n"))
  (print-header))

And here’s the table:

---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
  0       1       4       9      16      25      36      49      64      81
 10     121     144     169     196     225     256     289     324     361
 20     441     484     529     576     625     676     729     784     841
 30     961    1024    1089    1156    1225    1296    1369    1444    1521
 40    1681    1764    1849    1936    2025    2116    2209    2304    2401
 50    2601    2704    2809    2916    3025    3136    3249    3364    3481
 60    3721    3844    3969    4096    4225    4356    4489    4624    4761
 70    5041    5184    5329    5476    5625    5776    5929    6084    6241
 80    6561    6724    6889    7056    7225    7396    7569    7744    7921
 90    8281    8464    8649    8836    9025    9216    9409    9604    9801
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
100   10201   10404   10609   10816   11025   11236   11449   11664   11881
110   12321   12544   12769   12996   13225   13456   13689   13924   14161
120   14641   14884   15129   15376   15625   15876   16129   16384   16641
130   17161   17424   17689   17956   18225   18496   18769   19044   19321
140   19881   20164   20449   20736   21025   21316   21609   21904   22201
150   22801   23104   23409   23716   24025   24336   24649   24964   25281
160   25921   26244   26569   26896   27225   27556   27889   28224   28561
170   29241   29584   29929   30276   30625   30976   31329   31684   32041
180   32761   33124   33489   33856   34225   34596   34969   35344   35721
190   36481   36864   37249   37636   38025   38416   38809   39204   39601
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
200   40401   40804   41209   41616   42025   42436   42849   43264   43681
210   44521   44944   45369   45796   46225   46656   47089   47524   47961
220   48841   49284   49729   50176   50625   51076   51529   51984   52441
230   53361   53824   54289   54756   55225   55696   56169   56644   57121
240   58081   58564   59049   59536   60025   60516   61009   61504   62001
250   63001   63504   64009   64516   65025   65536   66049   66564   67081
260   68121   68644   69169   69696   70225   70756   71289   71824   72361
270   73441   73984   74529   75076   75625   76176   76729   77284   77841
280   78961   79524   80089   80656   81225   81796   82369   82944   83521
290   84681   85264   85849   86436   87025   87616   88209   88804   89401
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
300   90601   91204   91809   92416   93025   93636   94249   94864   95481
310   96721   97344   97969   98596   99225   99856  100489  101124  101761
320  103041  103684  104329  104976  105625  106276  106929  107584  108241
330  109561  110224  110889  111556  112225  112896  113569  114244  114921
340  116281  116964  117649  118336  119025  119716  120409  121104  121801
350  123201  123904  124609  125316  126025  126736  127449  128164  128881
360  130321  131044  131769  132496  133225  133956  134689  135424  136161
370  137641  138384  139129  139876  140625  141376  142129  142884  143641
380  145161  145924  146689  147456  148225  148996  149769  150544  151321
390  152881  153664  154449  155236  156025  156816  157609  158404  159201
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
400  160801  161604  162409  163216  164025  164836  165649  166464  167281
410  168921  169744  170569  171396  172225  173056  173889  174724  175561
420  177241  178084  178929  179776  180625  181476  182329  183184  184041
430  185761  186624  187489  188356  189225  190096  190969  191844  192721
440  194481  195364  196249  197136  198025  198916  199809  200704  201601
450  203401  204304  205209  206116  207025  207936  208849  209764  210681
460  212521  213444  214369  215296  216225  217156  218089  219024  219961
470  221841  222784  223729  224676  225625  226576  227529  228484  229441
480  231361  232324  233289  234256  235225  236196  237169  238144  239121
490  241081  242064  243049  244036  245025  246016  247009  248004  249001
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
500  251001  252004  253009  254016  255025  256036  257049  258064  259081
510  261121  262144  263169  264196  265225  266256  267289  268324  269361
520  271441  272484  273529  274576  275625  276676  277729  278784  279841
530  281961  283024  284089  285156  286225  287296  288369  289444  290521
540  292681  293764  294849  295936  297025  298116  299209  300304  301401
550  303601  304704  305809  306916  308025  309136  310249  311364  312481
560  314721  315844  316969  318096  319225  320356  321489  322624  323761
570  326041  327184  328329  329476  330625  331776  332929  334084  335241
580  337561  338724  339889  341056  342225  343396  344569  345744  346921
590  349281  350464  351649  352836  354025  355216  356409  357604  358801
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
600  361201  362404  363609  364816  366025  367236  368449  369664  370881
610  373321  374544  375769  376996  378225  379456  380689  381924  383161
620  385641  386884  388129  389376  390625  391876  393129  394384  395641
630  398161  399424  400689  401956  403225  404496  405769  407044  408321
640  410881  412164  413449  414736  416025  417316  418609  419904  421201
650  423801  425104  426409  427716  429025  430336  431649  432964  434281
660  436921  438244  439569  440896  442225  443556  444889  446224  447561
670  450241  451584  452929  454276  455625  456976  458329  459684  461041
680  463761  465124  466489  467856  469225  470596  471969  473344  474721
690  477481  478864  480249  481636  483025  484416  485809  487204  488601
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
700  491401  492804  494209  495616  497025  498436  499849  501264  502681
710  505521  506944  508369  509796  511225  512656  514089  515524  516961
720  519841  521284  522729  524176  525625  527076  528529  529984  531441
730  534361  535824  537289  538756  540225  541696  543169  544644  546121
740  549081  550564  552049  553536  555025  556516  558009  559504  561001
750  564001  565504  567009  568516  570025  571536  573049  574564  576081
760  579121  580644  582169  583696  585225  586756  588289  589824  591361
770  594441  595984  597529  599076  600625  602176  603729  605284  606841
780  609961  611524  613089  614656  616225  617796  619369  620944  622521
790  625681  627264  628849  630436  632025  633616  635209  636804  638401
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
800  641601  643204  644809  646416  648025  649636  651249  652864  654481
810  657721  659344  660969  662596  664225  665856  667489  669124  670761
820  674041  675684  677329  678976  680625  682276  683929  685584  687241
830  690561  692224  693889  695556  697225  698896  700569  702244  703921
840  707281  708964  710649  712336  714025  715716  717409  719104  720801
850  724201  725904  727609  729316  731025  732736  734449  736164  737881
860  741321  743044  744769  746496  748225  749956  751689  753424  755161
870  758641  760384  762129  763876  765625  767376  769129  770884  772641
880  776161  777924  779689  781456  783225  784996  786769  788544  790321
890  793881  795664  797449  799236  801025  802816  804609  806404  808201
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
900  811801  813604  815409  817216  819025  820836  822649  824464  826281
910  829921  831744  833569  835396  837225  839056  840889  842724  844561
920  848241  850084  851929  853776  855625  857476  859329  861184  863041
930  866761  868624  870489  872356  874225  876096  877969  879844  881721
940  885481  887364  889249  891136  893025  894916  896809  898704  900601
950  904401  906304  908209  910116  912025  913936  915849  917764  919681
960  923521  925444  927369  929296  931225  933156  935089  937024  938961
970  942841  944784  946729  948676  950625  952576  954529  956484  958441
980  962361  964324  966289  968256  970225  972196  974169  976144  978121
990  982081  984064  986049  988036  990025  992016  994009  996004  998001
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
          1       2       3       4       5       6       7       8       9
---  ------  ------  ------  ------  ------  ------  ------  ------  ------
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