## Puzzle| Amicable pairs (a,b) and (c,d) such that a + b = c + d

Amicable numbers
https://en.wikipedia.org/wiki/Amicable_numbers

$\sigma(n)$   is the sum of the positive divisors of   $n$,   including 1 and   $n$   itself
$s(n)$   is the sum of the proper divisors of   $n$,   which does not include   $n$   itself;
that is,   $s(n) = \sigma(n) - n$

The first pair is,    (220, 284)

$s(220) \; = \; \sigma(220) \; - \; 220 \; = \; 504 \; - \; 220 \; = \; 284$
$s(284) \; = \; \sigma(284) \; - \; 284 \; = \; 504 \; - \; 284 \; = \; 220$

Find amicable pairs   $(a, b)$   and   $(c, d)$   such that   $a + b = c + d$

Paul found:

(1162074914,   1232933086),     (1193945522,   1201062478)
(1273666394,   1408742566),     (1327395368,   1355013592)
(1558818261,   1596205611),     (1559592628,   1595431244)
(1741985325,   1857784275),     (1793313730,   1806455870)
(1938775850,   2110964950),     (1957374968,   2092365832)
(2141689095,   2574912249),     (2152573605,   2564027739)
(2223111345,   2383045455),     (2282361092,   2323795708)
(2419787565,   2675047635),     (2511601365,   2583233835)
(3274564293,   3623058747),     (3437861427,   3459761613)
(3396934125,   3427629075),     (3401192925,   3423370275)
(3892163756,   4033135444),     (3900906009,   4024393191)
(4023332270,   4250331730),     (4056852735,   4216811265)
(4617085725,   5410843875),     (4960585125,   5067344475)
(5301374925,   5894337075),     (5502767650,   5692944350)
(5513127795,   6585084045),     (5939023970,   6159187870)
(5574272535,   5841932265),     (5587409230,   5828795570)
(6682859001,   6790149639),     (6692989624,   6780019016)
(6885178846,   7593733154),     (6960968212,   7517943788)
(6943893957,   7397123643),     (7143425289,   7197592311)
(7317585748,   7740482732),     (7353700965,   7704367515)
(9535950765,   10390515795),     (9883587230,   10042879330)

(153015550,   168392450),     (156226856,   165181144
(165742395,   182622405),     (169335790,   179029010)
(273141836,   306014644),     (285627615,   293528865)
(275304590,   281534770),     (275631376,   281207984)
(305178874,   356714246),     (318580262,   343312858)
(311913855,   326754945),     (316293016,   322375784)
(359156770,   402020318),     (362645570,   398531518)
(393463072,   402877088),     (397287345,   399052815)
(427950350,   455921650),     (438452624,   445419376)
(540759375,   640415025),     (586188070,   594986330)
(621354896,   661063024),     (640001054,   642416866)
(965615992,   1102800008),     (989779882,   1078636118)

math grad - Interest: Number theory
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### 3 Responses to Puzzle| Amicable pairs (a,b) and (c,d) such that a + b = c + d

1. paul says:

Here are a few.

s(1162074914) = 2395008000 – 1162074914 = 1232933086
s(1232933086) = 2395008000 – 1232933086 = 1162074914

s(1193945522) = 2395008000 – 1193945522 = 1201062478
s(1201062478) = 2395008000 – 1201062478 = 1162074914

1162074914 + 1232933086 = 2395008000 &
1193945522 + 1201062478 = 2395008000

s(1273666394) = 2682408960 – 1273666394 = 1408742566
s(1408742566) = 2682408960 – 1408742566 = 1273666394

s(1327395368) = 2682408960 – 1327395368 = 1355013592
s(1355013592) = 2682408960 – 1355013592 = 1273666394

1273666394 + 1408742566 = 2682408960 &
1327395368 + 1355013592 = 2682408960

s(1558818261) = 3155023872 – 1558818261 = 1596205611
s(1596205611) = 3155023872 – 1596205611 = 1558818261

s(1559592628) = 3155023872 – 1559592628 = 1595431244
s(1595431244) = 3155023872 – 1595431244 = 1558818261

1558818261 + 1596205611 = 3155023872 &
1559592628 + 1595431244 = 3155023872

s(1741985325) = 3599769600 – 1741985325 = 1857784275
s(1857784275) = 3599769600 – 1857784275 = 1741985325

s(1793313730) = 3599769600 – 1793313730 = 1806455870
s(1806455870) = 3599769600 – 1806455870 = 1741985325

1741985325 + 1857784275 = 3599769600 &
1793313730 + 1806455870 = 3599769600

s(1938775850) = 4049740800 – 1938775850 = 2110964950
s(2110964950) = 4049740800 – 2110964950 = 1938775850

s(1957374968) = 4049740800 – 1957374968 = 2092365832
s(2092365832) = 4049740800 – 2092365832 = 1938775850

1938775850 + 2110964950 = 4049740800 &
1957374968 + 2092365832 = 4049740800

s(2141689095) = 4716601344 – 2141689095 = 2574912249
s(2574912249) = 4716601344 – 2574912249 = 2141689095

s(2152573605) = 4716601344 – 2152573605 = 2564027739
s(2564027739) = 4716601344 – 2564027739 = 2141689095

2141689095 + 2574912249 = 4716601344 &
2152573605 + 2564027739 = 4716601344

s(2223111345) = 4606156800 – 2223111345 = 2383045455
s(2383045455) = 4606156800 – 2383045455 = 2223111345

s(2282361092) = 4606156800 – 2282361092 = 2323795708
s(2323795708) = 4606156800 – 2323795708 = 2223111345

2223111345 + 2383045455 = 4606156800 &
2282361092 + 2323795708 = 4606156800

s(2419787565) = 5094835200 – 2419787565 = 2675047635
s(2675047635) = 5094835200 – 2675047635 = 2419787565

s(2511601365) = 5094835200 – 2511601365 = 2583233835
s(2583233835) = 5094835200 – 2583233835 = 2419787565

2419787565 + 2675047635 = 5094835200 &
2511601365 + 2583233835 = 5094835200

s(3274564293) = 6897623040 – 3274564293 = 3623058747
s(3623058747) = 6897623040 – 3623058747 = 3274564293

s(3437861427) = 6897623040 – 3437861427 = 3459761613
s(3459761613) = 6897623040 – 3459761613 = 3274564293

3274564293 + 3623058747 = 6897623040 &
3437861427 + 3459761613 = 6897623040

s(3396934125) = 6824563200 – 3396934125 = 3427629075
s(3427629075) = 6824563200 – 3427629075 = 3396934125

s(3401192925) = 6824563200 – 3401192925 = 3423370275
s(3423370275) = 6824563200 – 3423370275 = 3396934125

3396934125 + 3427629075 = 6824563200 &
3401192925 + 3423370275 = 6824563200

s(3892163756) = 7925299200 – 3892163756 = 4033135444
s(4033135444) = 7925299200 – 4033135444 = 3892163756

s(3900906009) = 7925299200 – 3900906009 = 4024393191
s(4024393191) = 7925299200 – 4024393191 = 3892163756

3892163756 + 4033135444 = 7925299200 &
3900906009 + 4024393191 = 7925299200

s(4023332270) = 8273664000 – 4023332270 = 4250331730
s(4250331730) = 8273664000 – 4250331730 = 4023332270

s(4056852735) = 8273664000 – 4056852735 = 4216811265
s(4216811265) = 8273664000 – 4216811265 = 4023332270

4023332270 + 4250331730 = 8273664000 &
4056852735 + 4216811265 = 8273664000

s(4617085725) = 10027929600 – 4617085725 = 5410843875
s(5410843875) = 10027929600 – 5410843875 = 4617085725

s(4960585125) = 10027929600 – 4960585125 = 5067344475
s(5067344475) = 10027929600 – 5067344475 = 4617085725

4617085725 + 5410843875 = 10027929600 &
4960585125 + 5067344475 = 10027929600

s(5301374925) = 11195712000 – 5301374925 = 5894337075
s(5894337075) = 11195712000 – 5894337075 = 5301374925

s(5502767650) = 11195712000 – 5502767650 = 5692944350
s(5692944350) = 11195712000 – 5692944350 = 5301374925

5301374925 + 5894337075 = 11195712000 &
5502767650 + 5692944350 = 11195712000

s(5513127795) = 12098211840 – 5513127795 = 6585084045
s(6585084045) = 12098211840 – 6585084045 = 5513127795

s(5939023970) = 12098211840 – 5939023970 = 6159187870
s(6159187870) = 12098211840 – 6159187870 = 5513127795

5513127795 + 6585084045 = 12098211840 &
5939023970 + 6159187870 = 12098211840

s(5574272535) = 11416204800 – 5574272535 = 5841932265
s(5841932265) = 11416204800 – 5841932265 = 5574272535

s(5587409230) = 11416204800 – 5587409230 = 5828795570
s(5828795570) = 11416204800 – 5828795570 = 5574272535

5574272535 + 5841932265 = 11416204800 &
5587409230 + 5828795570 = 11416204800

s(6682859001) = 13473008640 – 6682859001 = 6790149639
s(6790149639) = 13473008640 – 6790149639 = 6682859001

s(6692989624) = 13473008640 – 6692989624 = 6780019016
s(6780019016) = 13473008640 – 6780019016 = 6682859001

6682859001 + 6790149639 = 13473008640 &
6692989624 + 6780019016 = 13473008640

s(6885178846) = 14478912000 – 6885178846 = 7593733154
s(7593733154) = 14478912000 – 7593733154 = 6885178846

s(6960968212) = 14478912000 – 6960968212 = 7517943788
s(7517943788) = 14478912000 – 7517943788 = 6885178846

6885178846 + 7593733154 = 14478912000 &
6960968212 + 7517943788 = 14478912000

s(6943893957) = 14341017600 – 6943893957 = 7397123643
s(7397123643) = 14341017600 – 7397123643 = 6943893957

s(7143425289) = 14341017600 – 7143425289 = 7197592311
s(7197592311) = 14341017600 – 7197592311 = 6943893957

6943893957 + 7397123643 = 14341017600 &
7143425289 + 7197592311 = 14341017600

s(7317585748) = 15058068480 – 7317585748 = 7740482732
s(7740482732) = 15058068480 – 7740482732 = 7317585748

s(7353700965) = 15058068480 – 7353700965 = 7704367515
s(7704367515) = 15058068480 – 7704367515 = 7317585748

7317585748 + 7740482732 = 15058068480 &
7353700965 + 7704367515 = 15058068480

s(9535950765) = 19926466560 – 9535950765 = 10390515795
s(10390515795) = 19926466560 – 10390515795 = 9535950765

s(9883587230) = 19926466560 – 9883587230 = 10042879330
s(10042879330) = 19926466560 – 10042879330 = 9535950765

9535950765 + 10390515795 = 19926466560 &
9883587230 + 10042879330 = 19926466560

Paul.

2. paul says:

and a few more 9 digit ones

s(153015550) = 321408000 – 153015550 = 168392450
s(168392450) = 321408000 – 168392450 = 153015550

s(156226856) = 321408000 – 156226856 = 165181144
s(165181144) = 321408000 – 165181144 = 153015550

153015550 + 168392450 = 321408000 &
156226856 + 165181144 = 321408000

s(165742395) = 348364800 – 165742395 = 182622405
s(182622405) = 348364800 – 182622405 = 165742395

s(169335790) = 348364800 – 169335790 = 179029010
s(179029010) = 348364800 – 179029010 = 165742395

165742395 + 182622405 = 348364800 &
169335790 + 179029010 = 348364800

s(273141836) = 579156480 – 273141836 = 306014644
s(306014644) = 579156480 – 306014644 = 273141836

s(285627615) = 579156480 – 285627615 = 293528865
s(293528865) = 579156480 – 293528865 = 273141836

273141836 + 306014644 = 579156480 &
285627615 + 293528865 = 579156480

s(275304590) = 556839360 – 275304590 = 281534770
s(281534770) = 556839360 – 281534770 = 275304590

s(275631376) = 556839360 – 275631376 = 281207984
s(281207984) = 556839360 – 281207984 = 275304590

275304590 + 281534770 = 556839360 &
275631376 + 281207984 = 556839360

s(305178874) = 661893120 – 305178874 = 356714246
s(356714246) = 661893120 – 356714246 = 305178874

s(318580262) = 661893120 – 318580262 = 343312858
s(343312858) = 661893120 – 343312858 = 305178874

305178874 + 356714246 = 661893120 &
318580262 + 343312858 = 661893120

s(311913855) = 638668800 – 311913855 = 326754945
s(326754945) = 638668800 – 326754945 = 311913855

s(316293016) = 638668800 – 316293016 = 322375784
s(322375784) = 638668800 – 322375784 = 311913855

311913855 + 326754945 = 638668800 &
316293016 + 322375784 = 638668800

s(359156770) = 761177088 – 359156770 = 402020318
s(402020318) = 761177088 – 402020318 = 359156770

s(362645570) = 761177088 – 362645570 = 398531518
s(398531518) = 761177088 – 398531518 = 359156770

359156770 + 402020318 = 761177088 &
362645570 + 398531518 = 761177088

s(393463072) = 796340160 – 393463072 = 402877088
s(402877088) = 796340160 – 402877088 = 393463072

s(397287345) = 796340160 – 397287345 = 399052815
s(399052815) = 796340160 – 399052815 = 393463072

393463072 + 402877088 = 796340160 &
397287345 + 399052815 = 796340160

s(427950350) = 883872000 – 427950350 = 455921650
s(455921650) = 883872000 – 455921650 = 427950350

s(438452624) = 883872000 – 438452624 = 445419376
s(445419376) = 883872000 – 445419376 = 427950350

427950350 + 455921650 = 883872000 &
438452624 + 445419376 = 883872000

s(540759375) = 1181174400 – 540759375 = 640415025
s(640415025) = 1181174400 – 640415025 = 540759375

s(586188070) = 1181174400 – 586188070 = 594986330
s(594986330) = 1181174400 – 594986330 = 540759375

540759375 + 640415025 = 1181174400 &
586188070 + 594986330 = 1181174400

s(621354896) = 1282417920 – 621354896 = 661063024
s(661063024) = 1282417920 – 661063024 = 621354896

s(640001054) = 1282417920 – 640001054 = 642416866
s(642416866) = 1282417920 – 642416866 = 621354896

621354896 + 661063024 = 1282417920 &
640001054 + 642416866 = 1282417920

s(965615992) = 2068416000 – 965615992 = 1102800008
s(1102800008) = 2068416000 – 1102800008 = 965615992

s(989779882) = 2068416000 – 989779882 = 1078636118
s(1078636118) = 2068416000 – 1078636118 = 965615992

965615992 + 1102800008 = 2068416000 &
989779882 + 1078636118 = 2068416000

P.