Day: April 10, 2023

Number Theory

Number theory is a branch of math that explores the patterns and nature of positive whole numbers. We begin with introducing a concept termed amicable numbers, followed by a proof on finding amicable numbers, then turn to equivalent numbers, followed by an algorithm to obtain equivalent numbers.

4.1 Introduction to Amicable Numbers

The only application or use for these numbers is the original one – you insert a pair of amicable numbers into a pair of amulets, of which you wear one yourself and give the other to your beloved!

-John Conway

In the following sections, we build up to a complex proof for finding amicable numbers. Amicable (friendly) numbers are just as they sound! They are two numbers, say x and y, such that the sum of the divisors of x (except for x itself) will give you y, and the sum of the divisors of y (except for y itself) will give you x. So, for example, let x 220 let y = 284. The proper divisors of x are 1, 2, 5, 10, and 11, 20, 22, 44, 55, and 110. The proper divisors of y are 1, 2, 4, 71, and 142. So, =

and

1+2+5+10 +11+20 +22+44 +55 + 110 = 284

1+2+4+71 + 142 = 220.

Thus, 220 and 284 are amicable numbers.

We next introduce Thabit Ibn Qurra, who developed a rigorous proof for finding amicable numbers.

4.2 Thabit Ibn Qurra

Thabit Ibn Qurra was born in 836 CE during the Abbasid Caliphate in Harran, Upper Mesopotamia. Ibn Qurra grew up a Sabian, but later converted to Islam [14]. Ibn Qurra was persuaded by one of the three Banu-Musa brothers, who were all prolific mathematicians at the time, to go to Baghdad and learn from the Banu Musa brothers. Ibn Qurra was fluent in Syriac, Greek, and Arabic, and thus translated many works of Greek origin to Arabic. and he made discoveries in algebra, geometry, astronomy, physics, and number theory. He estimated the sidereal year as 365 days, 6 hours, 9 minutes, and 12 seconds. He overestimated the sidereal year by 3.5 seconds.

From the Ahl al Bayt none have ever expressed any doubt whatsoever about the Faith of Abu Talib. To the contrary all of them have, at some time or other, expressed their high opinions about his staunch Faith in the Prophet (s.a.) and Islam. This unanimity of opinion of the Ahl al bayt about Hazrat abu Talib is accepted by all the erudite scholars of Islam. Therefore Abul Ikram Abdus Salam ibne Mohammed says:

“The Imams of Ahl al Bayt are unanimous on the fact that Abu Talib died a Muslim. Whatever is contrary to the belief

of the Ahl al Bayt is never dependable.” Ref: Arhaj al Matalib, Page 268

Allama Tabarsi, of the Shia School of Thought writes: “Unanimity of the Ahl al bayt on the Faith of Abu Talib

is proven, and their unanimity is final and binding.”

Ref: Majma al Bayan, Vol 2, Page 287

Below are some of the sayings of the Prophet (s.a.) and the Infallible Imams (a.s.) which are the clear proof that they were all unanimous about the Faith of Abu Talib and his Deliverance on the Day of Reckoning.

“Abbas ibne Abd al Mutallib asked the Prophet (s.a.) if he expected the Deliverance of Abu Talib ? He said, ‘For him (Abu Talib) I expect Every good from Allah!””

Ref: Tabaqaat Ibne Saad, Vol 1, Page 124

Hazrat Ali ibne Abi Talib (a.s) said:

“Abu Talib didn’t die till he was sure that the Prophet (s.a) was completely

Happy and satisfied with him.”

Ref: Shara ibne Abil Hadeed, Vol 2, Page 312

Imam Zain al Abedeen (a.s.) was asked about the faith of Abu Talib, and he said:

“It is surprising that Allah had ordered the Prophet (s.a.) that no Muslim Woman be allowed to remain married to an infidel, and Fatima binte Asad

Who was one of the first ladies to embrace Islam

was Abu Talib’s spouse till her death.”

Ref: Shara ibne Abil Hadeed, Vol 2, Page 312

At this stage, this fact must be borne in minds that Fatima binte Asad embraced Islam during the early days of Baasat ( Annunciation) of the Prophet (s.a.), remained alive and the spouse of Abu Talib for ten years thereafter till her death. If they both belonged to different creeds, the natural consequence would have been religious dispute and separation. But no book reference about this..

of history history makes even a passing

Imam Mohammed Baqir (a.s.) has said:

“Abu Talib ibne Abd al Mutallib left this world

as a Momin (Muslim.)”

Ref: Al Hujjat ibne Saad, Page 27

Imam Jafar e Sadiq (a.s.) was told by a person that some persons were of the opinion that Abu Talib died an infidel. The Imam (a.s) said: “they are liars. Witnessing the prophethood of Mohammed (s.a.) Abu Talib has said: “Alam taalam ana wajadna muhammada Nabban la Moosa khat fi awwal al kitab “Don’t you know that we found Mohammed (s.a.) a Nabi the same way as Moosa (a.s.) whose references are there in the Earlier Books.” Ref: Usool e Kaafi, Page 244 Imam Moosa Kazim (a.s) was asked by Durust ibne Mansoor about the Faith of Abu Talib. The Imam (a.s.) replied: “He witnessed about the Prophet (s.a.) and everything that he (s.a.)



Brought!”

Ref: Usool e Kaafi, Page 242

Imam Reza (a.s.) wrote in a the reply to a letter from Abaan ibne Mehmood:

“If you don’t accept the Faith of Abu Talib, your

Journey will be in the direction of the Hell.”

Ref: Marat al Uqool, Vol 2, Page 264

Imam Hasan Askari (a.s.) has said:

“Abu Talib was like the Momins of the Aal e Firaun

who kept their Faith a closely guarded secret.

Ref: Al Hujjat ibne Saad, Page 115

In the early days after the Annunciation of the Prophet (s.a.), Abu Talib keeping his Faith secret and not openly expounding his beliefs before the infidels of the Qureish was an act of extreme discretion. If, immediately after the announcement of Mohammed (s.a.)’s Prophethood, he had declared his conversion to Islam, the infidels would have opened their skirmishes on two fronts. He wouldn’t have been able to help the Prophet (s.a.) by inviting the enmity of the Qureish against himself as well. Although it was not a secret to the infidels of the Qureish that at every occasion Abu Talib came to the rescue of the Prophet (s.a). Thus the voice of Islam was spreading and its circle was spreading. But they had no direct cause to find fault with Abu Talib and antagonize him. The statesmanlike attitude of Abu Talib provided him the opportunity to keep a sort of touch with the infidels and convey to them, in a subtle manner, the merits of Islam as a Faith. If Abu Talib had not adopted this attitude, the superior numbers of the Qureish at the time would have proved a more formidable foe for the fledgling group of Mohammed (s.a.) and his companions. Whatever opportunities the Prophet (s.a.) got to propagate his Faith was, to a great extent, due also to the covert and overt support of Abu Talib. If he was not there, the prevailing environment at Makka would have rendered Mohammed (s.a.)’s missionary work very slow. The well known scholar, Ibne Abil Hadeed who belonged to the Motazila Sect, acknowledged the contribution of Abu Talib to the spread of Islam thus:

“If Abu Talib and his son, Ali (a.s.) were not there, Islam Would never have stood on its feet,

One of them gave his support and help in Makka,

And the other put his own life in jeopardy in Madina”

This is a matter of great surprise that on the one hand it is acknowledged that Hazrat Abu Talib’s sacrifices and practical support helped the spread of Islam, and on the other they blatantly accuse him of infidelity. This opposition is from those who were willing to accept the weakest indications as evidence of Faith.and ardently supported certain persons who had a flipflop attitude of many a time thinking whether Mohammed (s.a.) was really a prophet or not! As opposed to this, they unreasonably count those persons out of Islam who valiantly supported the cause of Islam in its formative days. Abu Talib’s discreet silence at times and valiant support to his nephew at others was a great asset for the propagation of Islam. His practical life was moulded in the way of Islam. Every act of his was mirror


of the Prophet’s teachings and living proof of the veracity of Islam. He supported Islam, in word and in deed, with such zeal that was not possible for any person who was opposed to the creed that Mohammed (s.a.) was propagating.

Even if all these proofs and evidences are kept aside, no one can deny the profound love that Abu Talib had for the Prophet (s.a.). This great love for the Prophet (s.a.) itself is a strong proof of his acceptance of the creed. In the wildest imagination one cannot think that the love for the Prophet (s.a.) and the hate for his Creed could exist together in one person! If there is love for the Prophet (s.a.) in one’s heart, then naturally it cannot be devoid of love for Islam!

There does exist a group people who are mentally not prepared to endorse the infidelity of Abu Talib, but baseless, and false, traditions that have been surreptitiously introduced into the books of history cause them confusion. These traditions however do not qualify for veracity on the established yardstick for judging their correctness. They are the fabrications of the minions of the Umawi Court! Their attempt is to cover the services of Abu Talib to Mohammed (s.a.) and Islam because of their enmity with Hazrat Ali (a.s). It is necessary tomake an overview of the traditions concocted by them and people with vision can themselves read between the lines.

The first tradition says that when Abu Talib was on his death bed, the Prophet (s.a.) came near him. At that time Abu Jahl and Abd Allah ibne Omayya were there. The Prophet (s.a.) said, “Uncle! Recite’La ilaha il Allah’ so that I be witness of your Faith in Islam!” Abu Jahl and Abd Allah chided Abu Talib saying that if he did this, he would become an outcast from the tribe of Abd al Mutallib. Abu Talib now said,”I am from the Tribe of Abd al Mutallib!” and refused to recite the Kalima. The Prophet (s.a.) then said, “If I am not ordered (by Allah) to do otherwise, I shall continue to pray for your Deliverance!” At this juncture the following Verse was revealed:

“The Prophet and the Beleivers must ensure that they Don’t pray for the Deliverance of polytheists, even if they are their near ones, when it is evident to them that their destiny is The Hell.”

Now, the tradition quoted above is our subject of discussion on various counts:

Firstly, the narrator of this tradition is Musayyab. Ibne Hajar Asqalani writes in Tahzeeb al Tahzeeb that Musayyab is among the narrators who quote mainly from Abu Sufian ibne Harb and his own father Hazn, and he is quoted only by his son, Sayeed. It is evident therefore that neither his narrations were given any importance those days nor was he trusted by any one. There is another reason that he converted to Islam only at the time of the fall of Makka At the time of the demise of Hazrat Abu Talib neither was he present at his bed side nor had he access to any person of importance at the time. If he had heard of the incident from someone, the identity of the person was not established. Therefore the tradition is very weak and unacceptable. Besides this there was an element of bias in his reporting. The



person’s son, Saeed, sub-narrator, was a sworn enemy of Hazrat Ali( a.s.). Therefore, Ibne Abil Hadeed has written:

“Saeed ibne Musayyab was against Ali (a.s.) and opposed him.”

Ref: Shara Nahj al Balaga, Vil 1, Page 270

His enmity for the Ahl al Bayt is evident from the event of the demise of Imam Zain al Abedeen (a.s.) when his bier was brought to the Prophet’s Mosque in Madina and all the concourse of the town joined the Funeral Prayers, Sayeed continued to sit alone in the mosque and didn’t join the congregation. When asked why he didn’t join the prayer for that noble soul, he said:

“At this place of sanctity I prefer to offer two genuflections of prayer instead of joining the congregation for the Namaz e Janaza for that noble soul!”

Ref: Tabaqaat Ibne Saad, Vol 5, Page 222

He surpassed all limits of enmity that he blatantly refused to participate in the last prayers for the revered personage, remaining in the same surroundings all the while! Can the narration of such an inimical person be trusted to decide whether Hazrat Abu Talib was an infidel or not? His hatred for the progeny of Abu Talib has been proved without any element of doubt.

Secondly, this tradition is a contradiction of the tradition that Hazrat Abu Talib, in the throes of death, moved his lips and was reciting the kalmia of the Unity of Allah! This tradition has been quoted by many reputed historians in their books. Therefore, Abul Fida writes:

“When Abu Talib’s time of death approached, he moved his lips. Abbas took his ear near him and told to the Prophet, O! son Of my brother! Abu Talib has recited the Kalima that you wanted him to Recite. The prophet said: Thanks to Allah that He Guided You.”

Ref: Tareeq e Abul Fida, Vol 1, Page 120

INCIDENT OF COUNT GALARZA OF SPAIN

In addition to old devotees during the last stage of our saint’s life many Christians of higher status and wealthy gentlemen joined his fold and enjoyed the benevolence of his spiritual grace.

In December 1904 A.D. a young European gentleman came from Paris to meet Our Saint and bought with him an interpreter. After taking the oath of loyalty with extreme humility pleaded that he wanted to attain sublimity, see the Divine lustre and Divine manifestation 1n his life time, through the good offices of our Saint, Our Saint smilingly and kindly assuring, embraced him and gifted him with a few instructions consisting of secret and subtle points and from their special effects the fortunate person due to passionate love and excessive joy began to shed tears. Arrangements were made for his stay in Thakur Sahib’s house and Haji Aughat Shah was instructed to see his comforts.

The newly admitted devotee earn in evening to pay his respects and our Saint told the interpreter 10 make him understand God’s love cannot be obtained by rupees or gold coins. The one who does not mind his welfare or health, God meets him and if confirmation is required, in everything. His divine lustre is visible.

On the second day in the afternoon the saint sent for him and gave him sweets and semitahband with instructions to concentrate on one face and the same will be with him always. The employees or our Saint enquired the name of this god fearing Christian gentleman and came to know that he was a scion of an aristocratic family of Spain. His name was Count Galarza, son of a wealthy person. He was always camping in Paris in hotel continental for recreation. Besides Religious learning and other arts he was a doctor of Scholastic Philosophy.

Although he was pious and a believer of God’s Unity; he was in search of recluse and mystics. He came to know the renowned name and unique qualities of Our Saint from Ehtesham Ali Saheb who had gone to London for studies of a Barrister. From that time he was keen to meet Sarkar Waris. He spent one night in Dewa Sheriff and met three times and that too for liv minutes but acquired such spiritual knowledge in such a short period that others took days and days to grasp its significance and to comprehend its inner meaning.

A few days after the demise of Sarkar Waris, Count Galarza wrote a registered letter from Paris dated 13th May 1905 in which he added a paragraph for Haji Aughat Shah Saheb as follows. “I saw my honoured Saint going to the other world and at the same time of his death he fulfilled my desires and made me one with his heart”. From this it denotes his inner cleanliness and correct exposition. The mystics consider true revelation as symptoms of spirituality which after years of effort and endeavour is bestowed from the fountain head of bounty to the novice mystic seeking the path of truth. This appears a strange miracle of the bounties of Sarkar Waris that a Count sitting in Paris in Paris sees clearly the happenings of Dewa Sheriff. Thus he acquired the spiritual blessing from Our Saint beyond his expectations.

In addition to the above benefits he has also stated the other benefit he received and expresses his conviction that his Saint fulfilled all his desire as promised. What was his desire? It is already mentioned above, i.e. to witness the vision of Divine lustre. He also confessed that the Saint had bestowed this benefit is dignified and exalted and it is a combination of all his blessings.

This fortunate Christian had written a second letter dated 8th September 1905 to Haji Aughat Shah Sahib, mentioned only this minute study and stating that Sarkar Waris ordered those commandments in easy language which he had engraved on his heart. In his third letter dated 1st October 1905 he wrote to Aughat Shah in Arabic as his interpreter was away that he was free from literary activities and was diving in the sea of unity of God.

Count Galarza in metaphoric language expressed his spiritual activities in such eloquent language, the gist of which is an ambiguous picture of two sides from one side his spiritual dignity is apparent and from the other one side his research and ardour for unity, Although the two sides are concerned about the inner affair of unity, the difference is, the first one avers complete spirituality and the other the process of accomplishment.

One can infer that the count is absorbed in the Sea of Unity and is aware of the proofs of unity and in this way one can say that he is capable of observing and achieving his desire. The enquiry is not an ordinary achievement but pearl diving into the sea of unity is really a great dignity of the mystics.

For e.g. the well known mystic Sheikh Ali bin Ilahathi’s saying is, “Unity of God is superior to fields of knowledge”. And Abdu-r-Rahman Tafsunji states from all the learning, the knowledge or unity is superior.

Research Scholars among the mystics regarding the truth of the unity of God agreed that it is rather difficult to reach the vicinity of God. Only those with the acceptance of God, breaking their worldly connection, while passing through the valley solicitingly and willingly annihilate themselves before the Beloved, are ready to sacrifice their life to see the divine lustre are capable of achieving their goal. Beside these qualities and the attributes of Unitarians there are superior systems, mystics have also narrated. But, selected personalities are gifted and such sacred people come to the world after centuries and their exaltation and distinction is acknowledged even by strangers.

If anyone expressed doubt about Count Galarza’s external condition his achievement and exalted rank to the ordinary eye rather inconsistent. But in reality one should acknowledge he was always striving to know the real connection of unity.