bald chupacabras method...
for example, we take the square roots of the spaces and the keys themselves
pz65
2^64 √(18446744073709551616) 4294967296
30568377312064202855-18446744073709551616 12121633238354651239
√(12121633238354651239) 3481613596,933
2^63 √(9223372036854775808) 3037000499,976
4294967296/8 536870912
3481613596/8 435201699,5
536870912×x=3481613596 x = 6,485
3481613596/536870912 6,485
---
(4294967296/64)×x=(3481613596) x = 51,880085408687591552734375 pz 65
2^64 √(18446744073709551616) 4294967296
√(12121633238354651239) 3481613596,933 pz65
12121633231852051216 3481613596×3481613596
30568377305561602832 12121633231852051216+18446744073709551616 pz65
30568377312064202855 pz65
and divide into parts, by 2, by 3, by 64, 128, 1024, 2048, 4096, etc.
we catch such a divider into parts so that there are no repetitions
for 64 and 2 table
0 1
1 0
_______________1________________|_______________2_______________ 2
_______1_______|________2_______|_______3_______|_______4_______ 4
___1____|___2___|___3___|___4___|___5___|___6___|___7___|___8___ 8
|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_|_ 64
00000000000000000000010000000000 00000000000000000000000000000000 22 pz 12 0
00000000000000000000000000000000 00100000000000000000000000000000 35 pz 13 1
00000000000000000000000000000000 10000000000000000000000000000000 33 pz 14 1
00000000000000000000000000000000 01000000000000000000000000000000 34 pz 15 1
00000000000000000000000000000000 00000000000000000010000000000000 51 pz 16 1
00000000000000000000000000000000 00000000000000010000000000000000 48 pz 17 1
00000000000000000000000000000000 00000000001000000000000000000000 43 pz 18 1
00000000000000000000000000000000 00000000000010000000000000000000 45 pz 19 1
00000000000000000000000000000000 00000100000000000000000000000000 38 pz 20 1
00000000000000000000000000000000 00000000000000000010000000000000 51 pz 21 1
00000000000000000000000000000000 00000000000000000000010000000000 54 pz 22 1
00000000000000000000000000000000 00000000010000000000000000000000 42 pz 23 1
00000000000000000000000000000000 00000000000000000000010000000000 54 pz 24 1
00000000000000000000000000000000 00000000000000000000000000000010 63 pz 25 1
00000000000000000000000000000000 00000000000000000100000000000000 50 pz 26 1
00000000000000000000000000000000 00000000000000000001000000000000 52 pz 27 1
00000000000000000000000000000000 00000000000000000000100000000000 53 pz 28 1
00000000000000000000000000000000 00000000000100000000000000000000 44 pz 29 1
00000000000000000000000000000000 00000000000000000000000000001000 61 pz 30 1
00000000000000000000000000000000 00000000000000000000000000000100 62 pz 31 1
00000000000000000000000000000000 00000000010000000000000000000000 42 pz 32 1
00000000000000000000000000000000 00000000000000000001000000000000 52 pz 33 1
00000000000000000000000000000000 00000000000000000010000000000000 51 pz 34 1
00000000000000000000000001000000 00000000000000000000000000000000 26 pz 35 0
00000000000000000000000000000100 00000000000000000000000000000000 30 pz 36 0
00000000000000000000000000000000 00000000001000000000000000000000 43 pz 37 1
00000000000000010000000000000000 00000000000000000000000000000000 16 pz 38 0
00000000000000000000000001000000 00000000000000000000000000000000 26 pz 39 0
00000000000000000000000000000000 00000000000000000000000001000000 58 pz 40 1
00000000000000000000000000000000 00010000000000000000000000000000 36 pz 41 1
00000000000000000000000000000000 00010000000000000000000000000000 36 pz 42 1
00000000000000000000000000000000 00000000000000000001000000000000 52 pz 43 1
00000000000000000000000000000000 00000000000000000000001000000000 55 pz 44 1
00000000000000000000001000000000 00000000000000000000000000000000 23 pz 45 0
00000000000000000000000000000000 00000000001000000000000000000000 43 pz 46 1
00000000000000000000000000000000 00000000000000000000100000000000 53 pz 47 1
00000000000000000000000000000000 00000100000000000000000000000000 38 pz 48 1
00000000000000000000000000000000 00000000001000000000000000000000 43 pz 49 1
00000000000000000100000000000000 00000000000000000000000000000000 18 pz 50 0
00000000000000000000000000000000 00000000000000000000000001000000 58 pz 51 1
00000000000000000000000000000000 00000000000000000000000000100000 59 pz 52 1
00000000000000000000000000000000 00000000000010000000000000000000 45 pz 53 1
00000000000000000001000000000000 00000000000000000000000000000000 20 pz 54 0
00000000000000000000000000000000 00000000000000000001000000000000 52 pz 55 1
00000000000000000000000000000100 00000000000000000000000000000000 30 pz 56 0
00000000000000000000000000000000 00000000000000000000000000001000 61 pz 57 1
00000000000000000000000000000000 00000010000000000000000000000000 39 pz 58 1
00000000000000000000000000000000 00000000000000000000000001000000 58 pz 59 1
00000000000000000000000000000000 00000000000000000000000000000100 62 pz 60 1
00000000000000000000000000000010 00000000000000000000000000000000 31 pz 61 0
00000000000000000000000000000000 00000000000000000000100000000000 53 pz 62 1
00000000000000000000000000000000 00000000000000000000000000000100 62 pz 63 1
00000000000000000000000000000000 00000000000000000000000000001000 61 pz 64 1
00000000000000000000000000000000 00000000000000000010000000000000 51 pz 65 1
...............x.x.x.xx..x...xx. xxxx.xx..xxxx..xxxxxxxx.xxx.xxx. pz 66
00000000000000000000000000000000 00000100000000000000000000000000 38 pz 70 1
00000000000000000000000000000000 00000000000000000000000010000000 57 pz 75 1
00000000000000000000000001000000 00000000000000000000000000000000 26 pz 80 0
00000000000000000000000000000000 00000000000000000000000000001000 61 pz 85 1
00000000000000000000000000000000 00000000000000001000000000000000 49 pz 90 1
00000000000000000000000000000000 00000000000000000000100000000000 53 pz 95 1
00000000000000000000000000000000 00000000000000000100000000000000 50 pz 100 1
00000000000000000000000000000000 00000000000000010000000000000000 48 pz 105 1
00000000000000000000000000000000 00010000000000000000000000000000 36 pz 110 1
00000000000000000000000000000000 00000000000100000000000000000000 44 pz 115 1
pz 120
xxxx.xx..xxxx..xxxxxxxx.xxx.xxx. pz 125
for example, at 1664 repetitions stop falling out
1664 , 1664/2 832 ,1664/64 26, 26×x=1348 x = 51,846 pz 65
pz20
[572,924,858,887,1326,1250,1131,1190,1001,1337,1418,1096,1411,1645,1315,1360,1388,1168,1599,1628,1104,1353,1336,687,804,1127,438,701,
1511,950,936,1376,1442,615,1129,1392,996,1120,486,1514,1554,1178,545,1359,793,1594,1035,1508,1637,809,1387,1621,1604,1348,992,1494,
688,1587,1286,1403,1320,1251,941,1158]
35935527204195940 ~2^54
3+3+3+1+4+1+3+4+4+2+0+2+2+4+4+2+3+2+0+0+0+2+3+4+3+1+3+4+0+1+3+1 72 , 84-72 12,
3 3 3 1 4 1 3 4 4 2 0 2 2 4 4 2 3 2 0 0 0 2 3 4 3 1 3 4 0 1 3 1
x x x x . x x . . x x x x . . x x x x x x x x . x x x . x x x .
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
858 887 924 941 992 1035 1104 1120 1158 1178 1251 1286 1320 1337 1359 1403 1411 1442 1494 1508 1554 1587 1621 1645
936 996 1096 1129 1168 1190 1250 1315 1348 1376 1387 1418 1514 1604 1637
950 1001 1127 1336 1353 1392 1511 1594 1628
1131 1326 1360 1388 1599
1327 1482 1509 1534 1560 1586 1612 1638
1328 1483 1510 1535 1561 1588 1613 1639
1329 1484 1512 1536 1562 1589 1614 1640
1330 1485 1513 1537 1563 1590 1615 1641
1331 1486 1515 1538 1564 1591 1616 1642
1332 1487 1516 1539 1565 1592 1617 1643
1333 1488 1517 1540 1566 1593 1618 1644
1334 1489 1518 1541 1567 1595 1619 1646
1335 1490 1519 1542 1568 1596 1620 1647
1338 1491 1520 1543 1569 1597 1622 1648
1339 1492 1521 1544 1570 1598 1623 1649
1340 1493 1522 1545 1571 1600 1624 1650
1341 1495 1523 1546 1572 1601 1625 1651
1342 1496 1524 1547 1573 1602 1626 1652
1343 1497 1525 1548 1574 1603 1627 1653
1344 1498 1526 1549 1575 1605 1629 1654
1345 1499 1527 1550 1576 1606 1630 1655
1346 1500 1528 1551 1577 1607 1631 1656
1347 1501 1529 1552 1578 1608 1632 1657
1349 1502 1530 1553 1579 1609 1633 1658
1350 1503 1531 1555 1580 1610 1634 1659
1351 1504 1532 1556 1581 1611 1635 1660
1505 1533 1557 1582 1636 1661
1506 1558 1583 1662
1507 1559 1584 1663
1585
in general, everything is slipping somewhere, meaning that taking large divisors, we look in the table for 64 where they will fall out and so we select the spaces for the search.
the main thing is that more drops out on the right side than on the left if the table is divided into 2 equal sides, those that have already fallen out will not fall out, but you need to determine where exactly they can fall out, for example, there are parts that have not yet fallen out of 64, for example 37 40 41 46 47 56 60
we take the ones that haven't dropped yet and fit our search spots to them in the next puzzles
37 40 41 46 47 56 60 for 1664/2 832
833-1664
26×x=1348 x = 51,846 pz 65
962 37
963 37
964 37
965 37
966 37
967 37
968 37
969 37
970 37
971 37
972 37
973 37
974 37
975 37
976 37
977 37
978 37
979 37
980 37
981 37
982 37
983 37
984 37
985 37
986 37
987 37
1560 60
1561 60
1562 60
1563 60
1564 60
1565 60
1566 60
1567 60
1568 60
1569 60
1570 60
1571 60
1572 60
1573 60
1574 60
1575 60
1576 60
1577 60
1578 60
1579 60
1580 60
1581 60
1582 60
1583 60
1584 60
1585 60
etc
and so we are looking for where nothing has fallen out at all
pz67
(√(2^66)/2^30×536870912)^2+2^66 92233720368547758080
(√(2^66)/2^30×536870913)^2+2^66 92233720437267234880
92233720437267234880−92233720368547758080 68719476800 ~2^36
pz68
(√(2^67)/2^30×536870912)^2+2^67 184467440737095516160
(√(2^67)/2^30×536870913)^2+2^67 184467440874534469760
184467440874534469760-184467440737095516160 137438953600 ~2^37
pz69
(√(2^68)/2^30×536870912)^2+2^68 368934881474191032320
(√(2^68)/2^30×536870913)^2+2^68 368934881749068939520
368934881749068939520-368934881474191032320 274877907200 ~2^38
pz67
(√(2^66)/2^30×(2^30/2+0))^2+2^66 92233720368547758080
(√(2^66)/2^30×(2^30/2+1))^2+2^66 92233720437267234880
92233720437267234880−92233720368547758080 68719476800 ~2^36
pz68
(√(2^67)/2^31×(2^31/2+0))^2+2^67 184467440737095516160
(√(2^67)/2^31×(2^31/2+1))^2+2^67 184467440805814992928
184467440805814992928-184467440737095516160 68719476768 ~2^36
pz69
(√(2^68)/2^32×(2^32/2+0))^2+2^68 368934881474191032320
(√(2^68)/2^32×(2^32/2+1))^2+2^68 368934881542910509072
368934881542910509072-368934881474191032320 68719476752 ~2^36
***
pz65
(√(2^64)/1664×1348)^2+2^64 30552526466155465467,455
30568377312064202855
(√(2^64)/1664×1349)^2+2^64 30570494229757563437,443
30570494229757563437−30552526466155465467 17967763602097970 ~2^53
pz65
(√(2^64)/2048×1660)^2+2^64 30566001039707734016
30568377312064202855
(√(2^64)/2048×1661)^2+2^64 30580606952171110400
30580606952171110400−30566001039707734016 14605912463376384 ~2^53
pz66
(√(2^65)/2048×1660)^2+2^65 61132002079415468032
(√(2^65)/2048×1661)^2+2^65 61161213904342220800
61161213904342220800-61132002079415468032 29211824926752768 ~2^54
pz66
(√(2^65)/1664×1348)^2+2^65 61105052932310930934,911
(√(2^65)/1664×1349)^2+2^65 61140988459515126874,887
61140988459515126874-61105052932310930934 35935527204195940 ~2^54
18014398509481984 2^54
36028797018963968 2^55
pz66
(√(2^65)/4096×3320)^2+2^65 61132002079415468032
(√(2^65)/4096×3321)^2+2^65 61146605792855588864
61146605792855588864-61132002079415468032 14603713440120832
18014398509481984 2^54
pz67
(√(2^66)/2048×1660)^2+2^66 122264004158830936064
(√(2^66)/2048×1661)^2+2^66 122322427808684441600
122322427808684441600-122264004158830936064 58423649853505536 ~2^56
36028797018963968 2^55
(√(2^66)/4096×3320)^2+2^66 122264004158830936064
(√(2^66)/4096×3321)^2+2^66 122293211585711177728
122293211585711177728-122264004158830936064 29207426880241664 ~2^54
36028797018963968 2^55
pz68
(√(2^67)/2048×1660)^2+2^67 244528008317661872128
(√(2^67)/2048×1661)^2+2^67 244644855617368883200
244644855617368883200-244528008317661872128 116847299707011072 ~2^56
144115188075855872 2^57
pz68
(√(2^67)/4096×3320)^2+2^67 244528008317661872128
(√(2^67)/4096×3321)^2+2^67 244586423171422355456
244586423171422355456-244528008317661872128 58414853760483328 ~2^56
144115188075855872 2^57
pz69
(√(2^68)/2048×1660)^2+2^68 489056016635323744256
(√(2^68)/2048×1661)^2+2^68 489289711234737766400
489289711234737766400-489056016635323744256 233694599414022144 ~2^56
288230376151711744 2^58
36028797018963968 2^55
103864266406232064 3072 48-64 ((√(2^68)/3072×1661)^2+2^68)−((√(2^68)/3072×1660)^2+2^68)
82065593209862371,555 3456 54-64 ((√(2^68)/3456×1661)^2+2^68)−((√(2^68)/3456×1660)^2+2^68)
58423649853505536 4096 64-64 ((√(2^68)/4096×1661)^2+2^68)−((√(2^68)/4096×1660)^2+2^68)
pz69
(√(2^68)/4096×3320)^2+2^68 489056016635323744256
(√(2^68)/4096×3321)^2+2^68 489172846342844710912
489172846342844710912-489056016635323744256 116829707520966656 ~2^56
288230376151711744 2^58