How to remove query string page from the first page of laravel pagination?

The effective solution for this issue that I have found is to edit the pagination template.

First publish the pagination template from the vendors using the following command from the root of the project:

php artisan vendor:publish --tag=laravel-pagination

Now a file at resources/views/vendor/pagination/default.blade.php should be found and it could be edited like the following using str_replace() for the urls of each page and back navigation button:

<li><a href="{{str_replace('?page=1','',$paginator->previousPageUrl())}}" title="{{__('Previous')}}" rel="prev">{{$foxPrev}}</a></li>

and

 <li><a href="{{str_replace('?page=1','',$url)}}">{{ $page }}</a></li>

Update:

A bug was found with ?page=10 so instead of using str_replace we should using preg_replace like the following:

<li><a href="{{preg_replace('/\?page=[1]$/','',$url)}}">{{ $page }}</a></li>

Update 2:

In case of using any customized name for the page number parameter other than page, we could use the paginator getter for the $pageName property like the following in the pagination template:

<li><a href="{{preg_replace('/\?'.$paginator->getPageName().'=[1]$/','',$url)}}">{{ $page }}</a></li>

To know more about how to use more than one pagination on the same page or how to customize the page number parameter $pageName from view, checkout this answer

You can extend LengthAwarePaginator method url($page)

/**
 * Get the URL for a given page number.
 *
 * @param  int  $page
 * @return string
 */
public function url($page)
{
    if ($page <= 0) {
        $page = 1;
    }

    // If we have any extra query string key / value pairs that need to be added
    // onto the URL, we will put them in query string form and then attach it
    // to the URL. This allows for extra information like sortings storage.
    $parameters = ($page > 1) ? [$this->pageName => $page] : [];

    if (count($this->query) > 0) {
        $parameters = array_merge($this->query, $parameters);
    }

    return rtrim($this->path
        .(Str::contains($this->path, '?') ? '&' : '?')
        .http_build_query($parameters, '', '&')
        .$this->buildFragment(), '?');
}